A Set of Stabilizing PD Controllers

A Set of Stabilizing PD Controllers: An Application to Robot Manipulator

G. Leena, K. B. Datta & G. Ray

Journal of The Institution of Engineers (India): Series B Electrical, Electronics & Telecommunication and Computer Engineering ISSN 2250-2106 Volume 96 Number 1 J. Inst. Eng. India Ser. B (2015) 96:27-35 DOI 10.1007/s40031-014-0114-z

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Author's personal copy J. Inst. Eng. India Ser. B (January–March 2015) 96(1):27–35 DOI 10.1007/s40031-014-0114-z

ORIGINAL CONTRIBUTION

A Set of Stabilizing PD Controllers: An Application to Robot Manipulator G. Leena • K. B. Datta • G. Ray

Received: 19 October 2012 / Accepted: 20 February 2014 / Published online: 8 July 2014  The Institution of Engineers (India) 2014

Introduction

paper considers the problem of designing a class of stabilizing PD controllers for a robot manipulator. A major obstacle of designing the best controller has been the difficulty in characterizing the entire set of stabilizing controllers. Very few papers are published [1–3] where a class of stabilizing controllers is designed. An effective solution to this problem was obtained in [1]. This is accomplished by generalizing a classical stability result developed in the last century, the Hermite Biehler theorem. The characterization of all feedback gain values is useful for carrying out optimal designs with respect to various performance indices. In this communication, an attempt has been made to obtain a class of PD controllers based on PD stabilization theorem obtained from generalized Hermite Biehler theorem for each subsystem of MIMO nonlinear systems. The significant results of Siljak and Stipanovic [4] demonstrate how the linear matrix inequalities (LMIs) formulation can be used to quadratically stabilize linear/nonlinear interconnected system via centralized/decentralized linear constant feedback laws. Motivated by the work of Siljak and Stipanovic [4], the method by which the designed set of controllers for each subsystems can stabilize the MIMO nonlinear systems has been established.

One of the simplest position controllers for robot manipulator is the proportional derivative (PD) control. This

A Brief Description of all Stabilizing PD Controllers

G. Leena (&) Department of Electrical and Electronics, Faculty of Engineering and Technology, Manav Rachna International University, Sector 43, Aravali Hills, Faridabad, India e-mail: [email protected]

The feedback control system considered is shown in Fig. 1. Here r is the command signal; y is the output, G(s) = N(s)/ D(s) is the plant to be controlled; N(s) and D(s) are coprime polynomials. The controller C(s) is to be designed which is given by

Abstract This paper considers the problem of designing a class of stabilizing proportional derivative (PD) controllers for multi-input multi-output (MIMO) nonlinear systems. For a MIMO nonlinear system a set of stabilizing PD controllers were designed for each subsystem. The design approach is based on PD stabilization theorem derived from generalized result of the classical Hermite Biehler theorem. By considering the nonlinear terms as interconnections, a linear matrix inequalities optimization problem is formulated to ensure the stability of the composite nonlinear system with the designed decentralized controller parameters. A genetic algorithm based search technique is adopted to select an optimal PD controller gain from a search space of stabilizing controllers in order to have an optimum value of performance index. A two-link robot manipulator system is considered to show the effectiveness of the design procedure. Keywords Proportional derivative  Robot manipulator  Hermite Biehler theorem  Robust stabilizing controller

K. B. Datta  G. Ray Department of Electrical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, India

CðsÞ ¼ Kp þ sKd

ð1Þ

The closed-loop characteristic polynomial is

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r

J. Inst. Eng. India Ser. B (January–March 2015) 96(1):27–35

e

+

G(s) =

C(s)

pf ðx; Kp Þ ¼

N (s) D( s)

qf ðx; Kd Þ ¼

−

Fig. 1 Feedback control system

dðs; Kp ; Kd Þ ¼ DðsÞ þ ðKp þ sKd ÞNðsÞ

ð2Þ

The problem of stabilization using a PD controller is to determine the value of Kp, and Kd for which the closedloop characteristic polynomial d(s, Kp, Kd) is Hurwitz. A new polynomial is constructed by multiplying d(s, Kp, Kd) with N  ðsÞ [N  ðsÞ ¼ NðsÞ ¼ Ne ðs2 Þ  sNo ðs2 Þ; where Ne and No are the even and odd parts of N(s)] and examining the resulting polynomial, one can write rr ðdðs; Kp ; Kd ÞN  ðsÞÞ ¼ lðdðs; Kp ; Kd ÞN  ðsÞÞ  rðdðs; Kp ; Kd ÞN  ðsÞÞ ¼ lðdðs; Kp ; Kd ÞÞ  rðdðs; Kp ; Kd ÞÞ  ðlðNðsÞÞ  rðNðsÞÞÞ ð3Þ where l(d(s, Kp, Kd)) and r(d(s, Kp, Kd)) indicate the roots of characteristic polynomial d(s, Kp, Kd) in left and right half of complex plane respectively. Now, the closed-loop characteristic polynomial d(s, Kp, Kd), of degree n is Hurwitz if and only if l (d(s, Kp, Kd)) = n and r (d(s, Kp, Kd)) = 0. Equation (3) can be restated as 

rr ðdðs; Kp ; Kd ÞN ðsÞÞ ¼ n  ðlðNðsÞÞ  rðNðsÞÞÞ

ð4Þ

Determination those values of Kp, Kd is required for which (4) holds and d(s, Kp, Kd) N  ðsÞ has the following expression: ¼ ½s2 Do ðs2 ÞNo ðs2 Þ þ De ðs2 ÞNe ðs2 Þ þ Kp ðNe ðs2 ÞNe ðs2 Þ  s2 No ðs2 ÞNo ðs2 ÞÞ 2

2

ð1 þ x2 ÞðmþnÞ=2

ð5Þ

; :

It is to be seen from Eq. (6), that Kd appear in the odd part whereas Kp appears in the even part. Furthermore for every fixed Kp, the zeros of p(x, Kp) will not depend on Kd. The range of Kd for a fixed Kp can be obtained by solving the PD stabilization theorem. The PD stabilization theorem is derived from the results given in Ref. [1]. Further details on this theorem is shown in ‘‘Appendix’’. Our main contribution is to develop a set of stabilizing PD controllers for a class of MIMO nonlinear system and subsequently to investigate the stability of the controlled system based on LMI framework [5].

A Set of Stabilizing PD Controllers for a Class of Nonlinear System Let us consider a class of MIMO nonlinear system, which arises out of n interconnected subsystems, is described by x_i ¼ Ai xi þ Bi ui þ Bi hi ðt; xÞ;

i ¼ 1; 2; . . .; n;

yi ¼ Cni xi

ð7Þ

where xi are the states; ui, the inputs; and hi(t, x) are the nonlinear terms. The only information about the nonlinearity is that it satisfies the quadratic constraint. The transfer function of the ith decoupled system (without interaction terms, i.e. hi ðt; xÞ ¼ 0) is described by i ¼ 1; 2; . . .; n

ð8Þ

and a set of stabilizing PD controllers for each decoupled system was designed using the method described in [1].

2

þ s½Do ðs ÞNe ðs Þ  No ðs ÞDe ðs Þ

Stability Analysis

þ Kd ðNe ðs2 ÞNe ðs2 Þ  s2 No ðs2 ÞNo ðs2 ÞÞ

The nonlinear system (7) can be rewritten as

The new polynomial with s = jx is given by dðjx; Kp ; Kd ÞN  ðjxÞ ¼ pðx; Kp Þ þ jqðx; Kd Þ

ð6Þ

where p(x, Kp) = p1(x) ? Kp p2(x), q(x,Kd) = q1(x) ? Kd q2(x)   p1 ðxÞ ¼ De ðx2 ÞNe ðx2 Þ þ x2 Do ðx2 ÞNo ðx2 Þ ;   p2 ðxÞ ¼ Ne ðx2 ÞNe ðx2 Þ þ x2 No ðx2 ÞNo ðx2 Þ ;   q1 ðxÞ ¼ x Do ðx2 ÞNe ðx2 Þ  De ðx2 ÞNo ðx2 Þ ;   q2 ðxÞ ¼ x Ne ðx2 ÞNe ðx2 Þ þ x2 No ðx2 ÞNo ðx2 Þ : The new polynomial described by (6) is normalized in the following manner.

123

ð1 þ x2 ÞðmþnÞ=2 qðx; Kd Þ

Cni ðsI  Ai Þ1 Bi ;

dðs; Kp ; Kd ÞN  ðsÞ

2

pðx; Kp Þ

x_i ¼ Ai xi þ Bi ui þ wi ðt; xÞ yi ¼ Cni xi ;

ð9Þ

i ¼ 1; 2; . . .; n

where wi ¼ Bi hi : It is assumed that the pair (Ai, Bi) is stabilizable and the ith nonlinear term wi satisfies the quadratic constraints wTi ðt; xÞwi ðt; xÞ  a2i xT WiT Wi x;

i ¼ 1; 2; . . .; n

ð10Þ

where ai [ 0 are bounding parameters; and Wi are constant matrices of appropriate dimensions. Figure 2 shows the ith subsystem with the PD controller.

Author's personal copy J. Inst. Eng. India Ser. B (January–March 2015) 96(1):27–35

29

wi(t, x)(nonlinear term) refi + –

ith PD Controller

Bi

+

+

xi

∫

 αi C

Fig. 2 ith Subsystem with the PD controller for a nonlinear system

The input ui to the ith subsystem is ð11Þ

where ei = refi - yi is the error of ith subsystem, Kpi and Kdi are respectively, the proportional and derivative (PD) controller gains of the ith subsystem. Assume refi = 0, then ð12Þ

To study the stability of the interconnected system substitute for ui in Eq. (9) to get Eni x_i ¼ Ani xi þ wi ðxÞ

w ðxÞ

" XN   a2 W T Wi i¼1 i i

yi

Ai

ui ¼ Kpi ðyi Þ þ Kdi ðy_i Þ

T

0

+

ui ¼ Kpi ei ðtÞ þ Kdi e_i ðtÞ

x

T

ð13Þ

0

#"

x wðxÞ

I

# 0

ð15Þ

For the descriptor system (14), an augmented system was introduced to get the following equation.  þ wðxðtÞÞ  F z_ ¼ Az " #ð16Þ    x I 0 0 I where F ¼ ; A ¼ ; z¼ is 0 0 An En x_ " # 0  2n 9 1 vector and wðxÞ ¼ : wðxÞ A Lyapunov function candidate [6] is chosen for the descriptor system (16) as V ¼ zT FPz



P1 where P ¼ P2

0 P3

ð17Þ

 is non-singular with P1 ¼

PT1

[0

and P2, P3 [ 0 and FP ¼ ðFPÞT due to special structures of F and P. Then the following Eq. (18) is being simplified as  þw T ðxÞPz þ zT PT wðxÞ  V_ ¼ zT ðAT P þ PT AÞz

where Eni ¼ I þ Bi Kdi Cni ; i ¼ 1; 2; . . .; n

The descriptor system (16) is stable, provided the following conditions hold.

Ani ¼ Ai  Bi Kpi Cni ;

It may be noted that the set of PD decoupled controllers results the matrices Ani and Ei are in interval form. In a compact form the Eq. (13) can be rewritten as En x_ ¼ An x þ wðxÞ

P1 [ 0;

 þw T ðxÞPz þ zT PT wðxÞ\0  zT ðAT P þ PT AÞz ð18Þ

Equation (18) is equivalently written as

ð14Þ

where An = diag{An1, An2,…, Ann} and En = diag{En1, En2,…, Enn}are interval matrices of appropriate dimensions. In the compact notation, w = (wT1 , wT2 ,…, wTn )T and x = [xT1 , xT2 ,…, xTn ]T are the interconnection vector and the state vector respectively. A descriptor system approach to stability analysis is carried out [6] for the composite non-

P1 [ 0;

xT ðATn P2 þ PT2 An Þx þ x_T ðEnT P3  PT3 En Þx_

þ x_T ðP1  EnT P2 þ PT3 An Þx þ xT ðATn P3 þ P1  PT2 En Þx_ þ wT ðxÞP2 x þ wT ðxÞP3 x_ þ xT PT2 wðxÞ þ x_T PT3 wðxÞ\0 ð19Þ These inequalities can be rewritten as,

P1 [ 0; 2 T A P2 þ PT2 An  T T T 6 nT x x_ w ðxÞ 4 P3 An þ P1  EnT P2 P2

ATn P3 þ P1  PT2 En PT3 An þ P1  EnT P2

3 PT2 7 EnT P3  PT3 En PT3 5

P3

0

linear system while employing a set of decoupled PD controllers is discussed below. The constraint (10) is equivalent to the quadratic inequality [4]

2

x

3

6 7 6 x_ 7\0 4 5 wðxÞ

ð20Þ

By using S-procedure [7] it is possible to combine quadratic inequalities (15) and (20) into one single inequality (LMI) such that

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J. Inst. Eng. India Ser. B (January–March 2015) 96(1):27–35

2

P ATn P2 þ PT2 An þ b Ni¼1 a2i WiT Wi 4 PT3 An þ P1  EnT P2 P2

3 PT2 PT3 5\0 b I

ATn P3 þ P1  PT2 En EnT P3  PT3 En P3

where P1 [ 0 and a number b [ 0. By repeatedly applying the Schur-complement formula to Eq. (21) and b = 1, the Eq. (21) can be rewritten as

2

ATn P2 þ PT2 An 6 PT An þ P 1  E T P 2 n 6 3 6 P2 6 6 W1 6 6 .. 4 . Wn

ATn P3 þ P1  PT2 En EnT P3  PT3 En P3 0 .. .

PT2 PT3 I 0 .. .

W1T 0 0 c1 I .. .

0

0

0

    .. .

Arn1 T P2 þ PT2 Arn1 6 PT Ar 1 þ P1  E r 2 T P 2 n 6 3 n 6 P 2 6 6 W1 6 6 .. 4 . Wn

123

2

where r1 ; r2 ¼ 1; 2; . . .; 2n ; ci ¼ 1=a2i ;n is the size of the matrices An and En, i ¼ 1; 2; . . .; n:

3

WnT 0 0 0 .. .

7 7 7 7 7\0 7 7 5

ð22Þ

   cn I

where ci ¼ 1=a2i : The matrices An and En of Eq. (22) are interval matrices [obtained from Eq. (13)]. A sufficient condition for the stability robustness of interval matrices, i.e., matrices having the elements varying within given bounds, requires that the Lyapunov equation be negative definite when evaluated at the so-called corner matrices [8, 9]. The corner matrices of an n 9 n interval matrix A is defined as 2 Ar = {arij}, r ¼ 1; 2; . . .; 2n with arij = alij or auij, i, j = 1, 2,…, n, where alij and auij are minimum and maximum values respectively of ijth element of interval matrix. Hence Eq. (22) should be satisfied for all the corner matrices of An and En to ensure that the composite system (16) to be asymptotically stable. The matrix Wi is computed so that constraint (10) is satisfied and the bounding parameter ai is to be maximized. Hence Eq. (22) can be reformulated as an LMI optimization problem as stated below P Minimize Ni¼1 ci ; subject to P1 [ 0, and

2

ð21Þ

In other words, system (7) is robustly stabilized by the set of decoupled stabilizing PD controllers provided the LMI problem (23) has a feasible solution for all corner matrices.

Simulation Results The two-link manipulator has been considered as shown in Fig. 3 and its dynamics can be described by the nonlinear equation given below [10]. _ þ GðhÞ s ¼ MðhÞh€ þ Vðh; hÞ

ð24Þ

where M(h) is the 2 9 2 symmetric positive definite inertia _ is a 2 9 1 of Coriolis and centrifugal matrix; Vðh; hÞ vector; G(h) is a 2 9 1 gravity vector of the manipulator; h is the 2 9 1 vector representing joint angular positions; and s is the 2 9 1 vector of applied joint torques. The manipulator dynamic model (24) shows strong interactions between joint motions, since each element of M, N

Arn1 T P3 þ P1  PT2 Enr2 Enr2 T P3  PT3 Enr2 P3 0 .. .

PT2 PT3 I 0 .. .

W1T 0 0 c1 I .. .

    .. .

WnT 0 0 0 .. .

0

0

0



cn I

3 7 7 7 7 7\0 7 7 5

ð23Þ

Author's personal copy J. Inst. Eng. India Ser. B (January–March 2015) 96(1):27–35

31

and  G  can in general be a function of all joint variables h; h_ and M is a non-diagonal matrix, indicating inertial coupling between joint accelerations. For simplicity, it is denoted that _ ¼ Vðh; hÞ _ þ GðhÞ Nðh; hÞ

ð25Þ

_ due to There are uncertainties in M(h) and Nðh; hÞ unknown loads on the manipulator and unmodelled frictions. The following bounds are assumed on the uncertainties [11]: 1. 2.

There exist positive definite matrices Mu(h) and Ml(h) such that Ml ðhÞ  MðhÞ  Mu ðhÞ; _ and a nonnegative scalar function There exist Nu ðh; hÞ   _ _  Nðh; hÞ _   nmax ðh; hÞ _ nmax ðh; hÞ such that Nu ðh; hÞ

The state variables are defined to be x1 ¼ ½h1 h2 T ; x2 ¼ ½h_ 1 h_ 2 T and the concept of inverse dynamics of robot manipulator is utilized to have the control law in the form _ u ¼ Mu ðhÞ1 ðs  Nu ðh; hÞÞ

ð26Þ

Then x_1 ¼ h_ ¼ x2 ; _ x_2 ¼ h€ ¼ MðhÞ1 ðs  Nðh; hÞ _  Nðh; hÞÞ _ ¼ MðhÞ1 Mu ðhÞu þ MðhÞ1 ðNu ðh; hÞ ¼ Mðx1 Þ1 Mu ðx1 Þu þ Mðx1 Þ1 ðNu ðx1 ; x2 Þ  Nðx1 ; x2 ÞÞ ð27Þ The state space representation of Eq. (27) is represented by x_ ¼ Ax þ Bðu þ f ðxÞuÞ þ BhðxÞ

ð28Þ

where f ðxÞ ¼ Mðx1 Þ1 Mu ðx1 Þ  I; hðxÞ ¼ Mðx1 Þ1 ðNu ðx1 ; " # " # " # 0 I 0 x1 x2 Þ  Nðx1 ; x2 ÞÞ; A ¼ ;B ¼ and x ¼ : 0 0 I x2 Bf (x) is the uncertainty in the input matrix and in order to make its effect maximum f(x) is taken as f ðxÞ ¼ Ml ðx1 Þ1 Mu ðx1 Þ  I

ð29Þ

where Ml and Mu are the lower and upper bounds of inertia matrix which is found out as explained in [12]. In general, one can rewrite the Eqs. (24)–(29) for an n-link manipulator by considering h = [h1 h2]T and the basic structure of these equations are remain unchanged. After obtaining f(x) using Eq. (29), the combined system input matrix is transformed in decoupled form by considering f(x) as f ðxÞ ¼ diagðkmax ðf ð xÞÞ;

ð30Þ

which is obtained from the Rayleigh quotient lemma kmin ðf ÞxT x  xT k x  kmax ðf ÞxT x

ð31Þ

where kmin and kmax are minimum and maximum eigenvalues of f(x). The uncertainty h(x) has the following bounds [13]:

  khðxÞk ¼ Mðx1 Þ1 ðNu ðx1 ; x2 Þ  Nðx1 ; x2 ÞÞ    Mðx1 Þ1   kNu ðx1 ; x2 Þ  Nðx1 ; x2 ÞÞk

ð32Þ

1

 kM l ðx1 Þk ðnmax ðx1 ; x2 ÞÞ In many cases, the largest feasible region of x can be found out to determine a quadratic bound for khðxÞk2 such that hðxÞT hðxÞ  xT Qx

ð33Þ

where Q is a positive definite matrix. For the development of the decentralized control scheme, it is convenient to view each joint as a subsystem of the entire manipulator system and state variables are rearranged in Eq. (28) and are written as x_ ¼ An x þ Bn u þ BhðxÞ;

y ¼ Cn x

ð34Þ

where An ¼ diagfA1 ; A2 g; Bn ¼ B þ Bf ¼ diagfBn1 ; Bn2 g [f obtained from Eqs. (29) and (30)] and x ¼ ½x11 x12 x21 x22  T ¼ ½h1 h_ 1 h2 h_ 2 T : For the two-link robot considered, the matrices M(h), _ and G(h) are Vðh; hÞ   m11 m12 MðhÞ ¼ m12 m22   a1 þ a2 þ 2a3 cos h2 a2 þ a3 cos h2 ¼ ; a2 þ a3 cos h2 a2 2 3 2 ð35Þ  ða3 sin h2 Þðh_ 2 þ 2h_ 1 h_ 2 Þ _ ¼4 5; Vðh; hÞ 2 ða3 sin h2 Þh_ 1 " # a4 cos h1 þ a5 cos ðh1 þ h2 Þ GðhÞ ¼ a5 cos ðh1 þ h2 Þ In the above expressions a1, a2, …, a5 are constant parameters obtained from mass (m1, m2) and length (l1, l2) of robot links, that is, ½a1 ¼ ðm1 þ m2 Þl21 ; a2 ¼ m2 l22 ; a3 ¼ m2 l1 l2 ; a4 ¼ ðm1 þ m2 Þl1 g; a5 ¼ m2 l2 g: The parameters are m1 = m2 = 1.0 kg, l1 = l2 = 1.0 m and g = 9.81 m/ s 2. For the system (34), with the expressions (35) the following numerical values are obtained " # 0 1 A1 ¼ A2 ¼ ; 0 0 " # 0 Bn1 ¼ Bn2 ¼ ; 5:8 " # 0 ; B1 ¼ B2 ¼ 1 Cn1 ¼ Cn2 ¼ ½1

0;

and

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J. Inst. Eng. India Ser. B (January–March 2015) 96(1):27–35

" hðxÞ ¼

1:72x212 þ 1:68x222 þ 3:36x12 x22  ð5:12x212 þ 1:72x222 þ 3:44x12 x22 Þ

"

# ¼

h1 ðxÞ

#

m2

h2 ðxÞ ð36Þ

l2

A Set of PD Controllers for a Two-Link Robot Manipulator A set of decentralized PD controllers was designed for each of the two subsystems of (34) considering nonlinear interconnection term hi(x) as zero, as discussed in section ‘‘A Brief Description of all Stabilizing PD Controllers’’. The transfer function for each subsystem is given by Gi ðsÞ ¼ 5:8=s2 ;

i ¼ 1; 2

m1 l1

θ1

ð37Þ

The set of PD controllers for two subsystems is obtained as Kp1 [ 0; Kd1 [ 0 and Kp2 [ 0; Kd2 [ 0: From the class of stabilizing decentralized PD controllers a region is selected as Kp1 2 ½0:9 500; Kd1 2 ½0:9 100;

θ2

Fig. 3 Schematic of two-link revolute robot

ð38Þ

Kp2 2 ½0:9 500; Kd2 2 ½0:9 100

Suppose the desired positions for the robot is hd1 = 30, hd2 = 45. From the set of stabilizing controllers given in Eq. (38), optimal controller gains were found out using genetic algorithm based optimization technique [14] by maximizing fitness function 1 ; Jf ¼ 1þJ

J¼

Zt

e2i dt; i¼1; 2

ð39Þ

0

where ei ¼ hdi  hi ; hdi and hi are the desired and actual positions of joints 1 and 2 of the robot manipulator. The genetic operations applied were arithmetic crossover, uniform mutation and ranking selection. The population size of 80 was taken and GA was run for 25 generations. The optimal gain parameters obtained for hd1 = 30, hd2 = 45 are  Kp1 ¼ 466:28;

 Kd1 ¼ 48:68

 Kp2 ¼ 221:12;

 Kd2 ¼ 48:89

ð40Þ

Simulation results with the optimal gains given in Eq. (40) are shown in Figs. 4 and 5. The plots showing the desired position and actual position of joint 1 and 2 are sketched in Figs. 4 and 5, respectively. Stability Analysis of Two-Link Robot Manipulator The stability analysis of the two-link robot manipulator (34) was studied by solving the LMI optimization problem

123

Fig. 4 hd1 and h1 with optimal PD gains K*p1 = 466.28, K*d1 = 48.68, K*p2 = 221.12, K*d2 = 48.89

(23) for all the corner matrices of An and En. The designed ranges of An and En are calculated using Eq. (14) with the controller gains (38) and are given by An ¼ diagfAn1 ; An2 g;

ð41Þ En ¼ diagfEn1 ; En2 g     0 1 1 0 ; En1 ¼ ; where An1 ¼ ½2900  5:22 0 ½5:22 580 1     0 1 1 0 and En2 ¼ : An2 ¼ ½2900  5:22 0 ½5:22 580 1 As given by Eq. (10), the term wðt; xÞ; can be bounded by a quadratic inequality and is constrained as

Author's personal copy J. Inst. Eng. India Ser. B (January–March 2015) 96(1):27–35

33



2 wT1 ðt; xÞw1 ðt; xÞ ¼ 3:36x12 x22 þ 1:72x212 þ 1:68x222  17:07x12 x22  8:54ðx212 þ x222 Þ  xT a21 W1T W1 x;  29:44x12 x22  14:72ðx212 þ x222 Þ  xT a22 W2T W2 x ð42Þ (since x12 and x22 are much less than unity and the terms associated with the power of x12 and x22 equal to three or more than three are neglected), where a1 ; a2 [ 0 and W1 and W22are found out as 3 0 0 0 0 6 0 2:92 0 0 7 6 7 W1 ¼ 6 7 40 0 0 0 5 2

0 0

0 0

6 0 3:87 6 W2 ¼ 6 40 0 0

0

0 0 0

2:92 3 0 0 7 7 7 0 5

0

3:87

0

ð43Þ

Conclusions

Fig. 5 hd2 and h2 with optimal PD gains K*p1 = 466.28, K*d1 = 48.68, K*p2 = 221.12, K*d2 = 48.89 Table 1 Corner matrices of An and En A1n ¼ fAn1 ð2; 1Þ; An2 ð2; 1Þg

En1 ¼ fEn1 ð2; 1Þ; En2 ð2; 1Þg

A2n A3n A4n

¼ fAn1 ð2; 1Þ; An2 ð2; 1Þg

En2 ¼ fEn1 ð2; 1Þ; En2 ð2; 1Þg

¼ fAn1 ð2; 1Þ; An2 ð2; 1Þg

En3 ¼ fEn1 ð2; 1Þ; En2 ð2; 1Þg

¼ fAn1 ð2; 1Þ; An2 ð2; 1Þg

En4 ¼ fEn1 ð2; 1Þ; En2 ð2; 1Þg

Table 2 a1 ; a2 values obtained solving LMI problem with PD controller En1

En2

Two elements each of An and En are of interval form, i.e., four corner matrices for each of An and En are obtained. Table 1 shows the corner matrices of An and En. Thus there are sixteen combinations for which optimization problem (23) is solved using LMI control toolbox [15] with W1 ; W2 taken as Eq. (43). Table 2 shows the values of a1 ; a2 the bounding parameter values of the interconnection terms obtained by solving the LMI problem (23) with the designed range of controller parameters (30). It is seen that a feasible solution exists for all the combination of corner matrices. Hence, it is concluded that the set of decentralized PD controllers designed based on PD stabilization theorem stabilizes the two-link manipulator system (34). where Ani ð2; 1Þ andAni ð2; 1Þdenote the lower and upper limits of (2, 1)th element of matrix Ani :

En3

En4

A class of stabilizing PD controllers for a multi-input multi-output nonlinear system is designed based on generalized Hermite Biehler theorem. The nonlinear terms associated with the system dynamics were considered as interconnection terms. The set of stabilizing controllers is designed for the linear model (neglecting nonlinear interaction terms) and it is then proved that the designed set of PD controllers stabilizes the nonlinear system provided the nonlinearity term satisfies certain quadratic constraints and subsequently, the feasible solution of bounding parameters (a) for the nonlinear terms exits. The proposed decentralized information structure constraints ensure connective stability of the composite system while a set of decentralized robust stabilizing controller is adopted for a class of nonlinear interconnected systems. The genetic algorithm based optimization technique is employed to get an optimal controller gain from the range of stabilizing controller parameters. Simulation results are given to illustrate the efficacy of the proposed set of PD controller for a two-link robot manipulator system.

Appendix: PD Stabilization Theorem With a fixed Kp, the gain Kd is solvable for a plant with transfer function G(s) if and only if the following conditions hold: 1.

a1

a2

a1

a2

a1

a2

a1

a2

A1n

0.0216

0.0188

0.0216

0.0188

0.0216

0.0188

0.2225

0.1933

A2n

0.0216

0.0188

0.0216

0.0188

0.1973

0.1714

0.2228

0.1953

A3n

0.1970

0.1712

0.1973

0.1714

0.1973

0.1714

0.2246

0.1950

A4n

0.0216

0.0188

0.1973

0.1714

0.0216

0.0188

0.2228

0.1935

FK p is not empty i.e., at least one feasible string exists and FK p is given as

FK p ¼ AKp ðn  ðlðNðsÞÞ  rðNðsÞÞÞÞ

ð44Þ

The set AKp is defined as a set of all possible strings of 1’s, 0’s and -1’s whose length is l or l ? 1 depending on

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Author's personal copy 34

J. Inst. Eng. India Ser. B (January–March 2015) 96(1):27–35

x NðjxÞ2 [ 0 since N  ðsÞ does not have any zeros on jx axis. Hence

the value of n ? m (even or odd) for three possible conditions of N  ðsÞ: If N  ðjxt Þ ¼ 0 for some t = 1, 2, …, (l - 1), then define it ¼ 0; (b) If N  ðsÞ has a zero of multiplicity k at the origin, then define

(a)

q1 ðxj Þ \Kd q2 ðxj Þ

if

ij q2 ðxj Þ [ 0

ð47Þ

ðkÞ

i0 ¼ sgn½q1f ð0Þ where q1f ðxÞ ¼ (c)

;

For all other N ðjxt Þ 6¼ 0; t = 0, 1, 2, …, l, it 2 f1; 1g 

ffi0 ; i1 ; . . .; il1 g for ffi0 ; i1 ; . . .; il g for

A Kp ¼

2.

(b) q1 ðxÞ ð1þx2 ÞðmþnÞ=2 

n þ m eveng for n þ m oddg

ð45Þ

There exists a string R ¼ fi0 ;i1 ;...g 2 FK p such that h i h i 1 1 Re  maxit 2R;it [0 Re  jxt Gðjx \min i 2R;i \0 t t jxt Gðjxt Þ tÞ

where for a given fixed Kp, let 0\x1 \x2 \ x3 \    \xl1 be the real, nonnegative, distinct finite zeros of pf (x, Kp) with odd multiplicities (it is to be noted that these zeros are independent of Kd). Also, if the feasible strings satisfy the above conditionR1 ; R2 ; . . .; Rs 2 FK p ; then the set of all stabilizing gains is given by s

Kd ¼ [ Kr

ð46Þ

r¼1

where 0 B B Kr ¼ B @



 1 max Re  ; it 2Rs ;it [ 0 jxt Gðjxt Þ   1 min Re  it 2Rs ;it \0 jxt Gðxt Þ

1 C C C; A

r ¼ 1; 2; . . .; s

Proof From Eq. (4), it is found that dðs; Kp ; Kd Þ is Hurwitz if and only if R 2 AKp ðn  ðlðNðsÞÞ  rðNðsÞÞÞÞ ðkÞ sgn½qf ð0; Kd Þ; ij

¼ sgn½qf where R ¼ fi0 ; i1 ; . . .g; i0 ¼ ðxj ; Kd Þ and j = 1, 2, …, (l - 1) or j = 1, 2, …, l, accordingly as (m ? n) is even or odd. Now consider two different cases: Case 1: N  ðsÞ does not have any zeros on the imaginary axis: In this case for all stabilizing values of the gain Kd ; dðs; Kp ; Kd ÞN  ðsÞ will also not have any zeros on the jx axis so that ij 2 f1; 1g for j = 1, 2, …, l. Two different possibilities are considered below. (a)

If ij [ 0, then the stability requirement is q1 ðxj Þ þ Kd q2 ðxj Þ [ 0: From Eq. (44), it is noted that q2 ðxÞ ¼

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If ij \ 0, then the stability requirement is q1 ðxj Þ þ Kd q2 ðxj Þ\0: According to the same explanation as given in (a) and subsequently, q2 ðxÞ [ 0; it follows that

q1 ðxj Þ [ Kd q2 ðxj Þ

ð48Þ

Case 2: N  ðsÞ has one or more zeros on the jx axis including a zero of multiplicity k at the origin. In this case for all stabilizing values of the gain Kd ; dðs; Kp ; Kd ÞN  ðsÞ will also have the same set of jx axis zeros. Furthermore, it is clear that these zero locations will be a subset of fx1 ; x2 ; . . .; xl1 g: Since the location of these zeros depends on N  ðsÞ and is independent of the gain Kd, it is reasonable to expect that such a zero, at xm say, will not impose any additional constraint on Kd. Instead, it will constrain im 2 R to a particular value. Those conditions have been incorporated in the definition of AKp [(a) and (b) of Eq. (45)]. Of the two cases discussed above, only case 1 imposes constraints on Kd as given by Eqs. (48) and (49). Thus it is concluded that each ij [ 0 in the string R 2 AKp ðn  ðlðNðsÞÞ  rðNðsÞÞÞÞ contributes a lower bound on Kd while ij \0 contributes to the upper bound on Kd. For a fixed Kp, in order that the string R 2 AKp ðn  ðlðNðsÞÞ  rðNðsÞÞÞÞ corresponds to a stabilizing Kd, the condition to be satisfied is     q1 ðxt Þ q1 ðxt Þ max \ min ð49Þ it 2R; it [ 0 q2 ðxt Þ it 2R; it \0 q2 ðxt Þ 2

2

Ne ðs ÞþsNo ðs Þ Now, GðsÞ ¼ NðsÞ DðsÞ ¼ De ðs2 ÞþsDo ðs2 Þ ; so that

1 De ðx2 Þ þ j x Do ðx2 Þ ¼ jxðGðjxÞÞ jx ½Ne ðx2 Þ þ j x No ðx2 Þ q1 ðxÞ þ jp1 ðxÞ ¼ q2 ðxÞ Hence it follows that   q1 ðxÞ 1  ¼ Re  q2 ðxÞ jxt ðGðjxt ÞÞ

ð50Þ

h i 1 Thus, from hEq. (49),i maxit 2R; it [ 0 Re  jxt Gðjx tÞ 1 \ minit 2R; it \0 Re  jxt Gðjx : tÞ

Author's personal copy J. Inst. Eng. India Ser. B (January–March 2015) 96(1):27–35

This is condition (2) in the statement of the theorem. This completes the proof of the necessary and sufficient conditions for the existence of a stabilizing Kd. The set of all stabilizing Kd’s is now determined by taking the union of all Kd’s that are obtained from all feasible strings which satisfy (2).

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