Design of Nonlinear State Feedback Control

Law for Underactuated TORA System:

A Block Backstepping Approach Shubhobrata Rudra, Ranjit Kumar Barai, Madhubanti Maitra, Dharmadas Mandai, Shimul Dam, Somsubhra Ghosh,

Prarthna Bhattacharyya and Arka Dutta Electrical Engineering Department, Jadavpur University, Kolkata-700032, India

Abstract-This paper presents the formulation of a novel block-backstepping based control algorithm to address the stabilization problem for the well-known nonlinear benchmark TORA system. The ideas behind the method are as follows. At first, state model of the TORA system has been converted into block-strict feedback form. Then the control Lyapunov function has been designed for each cascaded dynamic block to derive the expression of the control input for the overall nonlinear system. The overall asymptotic stability of the TORA system has been analyzed using Lyapunov Stability Criteria. Finally, the effectiveness of the proposed control algorithm has been verified in the simulation environment. Keywords: Underactuated System, TORA System, Block Backstepping Control, Lyapunov Stability Criteria.

I. INTRODUCTION

Stabilization of a TORA system (Translational Oscillator with a Rotational Actuator) has been considered as an active research area for control system engineers [1-4]. The TORA system has been considered as a benchmark nonlinear system for the class of underactuated mechanical systems. Nowadays, TORA system has received a conspicuous amount of research attention [1]-[4]. Early research activities on TORA were initiated by the need of the design of the controllers to address the complex control problem of a dual-spin aircraft, where the interaction between spin and nutation complicates the algorithm design task for the control system engineers. Out-of-the-way, controlling the TORA is of independent interest as a benchmark problem in nonlinear control design [4]. Consequently, devising an efficient control algorithm for a TORA system remains as an active area of research. However, the control design of a TORA system is more complicated than that of a fully actuated system. Moreover, complicated coupling action between translational motion of the cart and angular rotation of the eccentric mass make the control design task more complicated.

Backstepping is Lyapunov method based versatile control design approach for nonlinear systems that ensures the convergence of the regulated variables to zero [5-7]. Backstepping relies on the fact that the system under

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consideration should be in a strict feedback form. Generally, single input single output systems satisfy this condition under some simplifying assumptions [5-9]. Nevertheless, in case of multi variable control problem, quite often the system structure is not in the lower triangular form (strict feedback form or semi strict feedback form). In fact, ordinary backstepping eventually fails to generate a control algorithm for MIMO systems. However, if it is possible to represent the state space structure of such system in block strict feedback form by means of some state transformation, it is possible to address the control problems of MIMO system using backstepping technique. This technique is also known as block backstepping [8-10].

The seminal contribution of Reza Olfati Saber in the field of underactuated system's research makes it possible to formulate different types of control algorithm for stabilization problem of the underactuated system [11-19]. Moreover, it is an inevitable fact that the proposed transformation [21] of an underactuated system's state model in the normal form simplifies the design task for the control system engineers. However, his proposed backstepping based control law [described in 11-13] is somewhat complicated and design of control algorithm for a TORA system using normal form is not a very convenient way to design a backstepping controller for the TORA system.

The main objective of the paper is to formulate a block backstepping based control algorithm for the TORA system. To accomplish the mentioned objective the paper is organized as follows: section II describes the mathematical model of the TORA system and precisely formulates the control objective in an analytic manner. Section III describes the derivation of the proposed algorithm. Section IV Analyzes the stability of the proposed control law. Section V describes the implantation procedure of the proposed algorithm on TORA system and analyzes the results. Finally, section V concludes the work.

II. PROBLEM FORMULATION

The nonlinear TORA system considered in this section was introduced by Wan, Bernstein and Coppola in [4]. The picture in Fig. 1 illustrates the top view of this nonlinear benchmark mechanical system in which the rotational

94 Proceedings of7'h International Conference on Intelligent Systems and Control (ISCO 2013)

motion of an eccentric mass controls the translational oscillations of the platform. Assuming that the platform moves in the horizontal plane, the dynamics of the system are described by

FIG. I: TORA SYSTEM

The Lagrangian model of the TORA system has been shown below [19]:

m"q, +m, A2 +� (q,q) = 0

m2,q, +m2A2 +hz (q,q) = r

where

mil =M+m m'2 = m2, = mrcosfJ m22 = 1 +mr2

� = -mr sin fJ(i + kx hz = -m2gr sin q2 Ll = mil m22 -m'2m2'

Applying the following feedback law:

: -m:r2 sin(2q2)p�+kq,m2rcos(q2)

-m2grsin(q2)(m, +m2)= u

. . . (1)

. . . (2)

One may able to realize the following state model structure for underactuated mechanical system [19].

q, =p, p, =-k3q, +k2 Sin(q2)p,2 -k2 COS(q2)U q2 = P2 P2 =U

where

k2 = m2r /( m, + m2) k3 = -k,/(m, +m2)

. . . (3)

The goal of the control objective is to ensure the asymptotic stabilization of the TORA system. That is, the control algorithm is required to ensure the asymptotic convergence of all the state variables towards their equilibrium.

III. DERIVATION OF THE CONTROL LAW

In this section, a generalized control law based on block backstepping has been proposed to address the stabilization problem of underactuated mechanical system. The steps of the control algorithm design are as follows:

Step 1: Define the new control variable z\ as follows

z,=q,-k(qz+g, Pz-gzp,) . . . (4)

where, k is a design constant.

The dynamics of z\ is expressed in equation (3)

z, = p,-k(pz +g,P2 +g,p, -g2P,-g2P,) = p, -k(p, + g, (I, + g2U)+ P2dg,.Q-g, (J; + g,u)-p,dg,.Q) = p, -k (p, + gJ2 + p,dg,.P -g2J; -p,dg2 .p)

. . . (5)

In above representation Q = [q, q, r ,P = [PI P2 r dg = [� �] and for ease of representation J.. ( q, p),

8q, 8q,

J;(q,p) , g, (q), g2(q)are represented by J..,J;,g"g2 respectively.

Step 2: Choice of stabilizing function for z\ subsystem. A conventional choice for stabilizing function is given below,

a, = -c,z, +k(P2 + g,J; + P2dg,.P-g2J.. -p,dg2·P )

Step 3: the second error variable is defined as follows

z2=p,-a, ... (6) with the above definition of second error variable, the dynamics of first error variable becomes:

. . . (7)

Dynamics of second error variable Z2 can be expressed as

i2 = p, -eX, =J.. + g,u + c,i, -k[{P2 + fsJ2 + gJ; + P2dg,.p

= h +g\u+q (Z2 -qz\) -k[{.I; + g2u+dg\ .p.l; + g\%.X +(.1; + g2u)dg\.p

+ P2 (ag\ (h + g\u) + 8g\ (.I; + g2U)J 8q1 8q2

Design of Nonlinear State Feedback Control Law for Underactuated TORA System: A Block Backstepping Approach 95

= ljIu + +c1 (Z2 -c1 ZI ) + ¢

where,

. . . (8)

Step 4: the control law u has been designed to ensure the desired dynamics for Z2. The desired dynamics of Z2 is expressed in equation (7)

. . . (9)

Consequently, the desired control input is given by following equation (9) as follows:

u=IjI-' [-( 1-cn z,- (c,+c2 ) z2-¢] . . . (10)

IV. PROOF OF STABILITY

Lemma: Suppose that an underactuated system which can be represented by the state model of equation (1). Then there exists a global state transformation for the system as described by equations (2)-(5), which transforms the system into a block strict feedback form. Moreover, there exists a state feedback law, which ensures the semi-global stabilization of the equilibriwn for the underactuated system.

Proof: the fust part of the Lemma (i.e. the transformation of the underactuated system in block strict feedback form) can be proved by direct calculation the stability property of the control law has been analyzed in the following section.

The proposed control law of equation (8) transforms the closed loop system in following forms:

2, = Z2 -C,Z, 22 = -ZI -C2Z2

. . . (11)

Now the following Lyapunov function has been defined for the transformed system described by equation (10) as follows:

1 2 1 2 V =-z +-z 2

1 2

2 . . . (12)

The derivative of the Lyapunov Function can be computed as follows:

V = z, (Z2 -C,ZI ) + Z2 (ZI -C2Z2 ) . . . (13)

The negative-defmiteness of V as expressed in equation

(11) implies the fact that V ( t) � V (O) and therefore it

ensures the boundedness of z, and Z2.

96 Proceedings of7'h International Conference on Intelligent Systems and Control (ISCO 2013)

Now define the following new function:

N(t) = C1Z� +C2Z� ... ( 14)

Integration of (12) results

{

V(t) = V(ZI (0), Z2 (0))+ fV(r)dr o ... ( 15)

{

= V(ZI (0),Z2 (0))+ fN(r)dr o

Thus

{

fN(r)dr = V(ZI (0),Z2 (O))-V(t) ... ( 16)

Considering V (t) � 0 and V (t) � 0 the following results

can be easily derived that

{

lim fN( r)dr � 00 {-"'O

o

The derivative of V can be expressed as

V = [2c,z,z, + 2CIZ2Z2]

... ( 17)

. . . ( 1 8) Since z" Z2' Z" Z2 are bounded, Equation ( 16) implies the

fact that V (t ) is a continuous function of time. Hence with

the application of Barbalat's Lemma it can be proved that z,

and Z2 converges to zero as t � 00 .

V. SIMULATION RESULTS

Comparing the state model of the TORA system with standard underactuated state model, the following expressions for fl' f2' gl and g2 can be obtained as follows:

1; (X) = -k3ql + k2 sin (q2) PI2

h (X)=O gl (X) = -k2 cos (qJ g2 (X)= 1

Now, define z, as follows:

ZI = ql -k (qz + PI +k2 COSq2P2 ) The derivative of z, is computed as follows:

The stabilizing function has been defined as:

al = -CIZI + k(P2 -k3ql)

Now second error variable Z2 is defined as

... ( 19)

... (20)

... (21)

... (22)

... (23)

The derivative of Zz becomes:

Z2 = -u ( k + k2 cos q2) + ci ( Z2 -ci ZI )

+ k [ -k,ql +k2 sin q2P; + kk,P, ] The expression of the control law has similar to equation (8) as follows:

u = Ij/-I [-(l-ct)z, -(ci +C2)Z2 _¢ ] where

Ij/-I = -1/1 + kk2 cos ( q2) and

¢ = [-k,q, + k2 sin q2P; + kk,p,]

. . . (24 )

been expressed

. . . (25 )

The control input has been applied on a TORA system, where the parameters of the TORA system are listed in Table 1. The simulation has been carried out for 1 00 seconds (with

following initial condition: (CiI =o.s,(]z =Ji16 and PI = P2 = 0). It has been observed that the state of the system converges to their desired equilibrium position in state space. The state of the under ctuated system has been shown in Figures 2-5 . In the mentioned simulation following values of the controller, parameters have been used: c,=2, C2 = 2 and k = 0 .56 .

TABLE 1: PHYSICAL PARAMETERS OF THE TORA SYSTEM (SI UNITS)

Cart Mass (M)

Spring Constant K

0.8

0.6

04

0.2

·0.2

-0.4

-0.6

-O.B

II

2.7

300

0.1

Pendulum Mass 0.2

Length (r) 0.18

12 0.148

-10 10 20 30 40 50 60 70 80 90 100 time(sec)-->

FIG. 2: V ARIA nON OF CART DISPLACEMENT WITH TIME

0.4

0.3

0.2

0.1

-0.1

-0.2

-0.3

-0.4

-O .50e-----,,�0 -----'O2� 0 -----'O3�0 -----'4'=-0 -----'5'=-0 -----"'=-0 -----'7'=-0 -----" '=-0 -----'9'= -0 -----c-!'00 Time(sec}------>

FIG. 3: V ARIA nON OF CART VELOCITY WITH TIME

Design of Nonlinear State Feedback Control Law for Underactuated TORA System: A Block Backstepping Approach 97

0.8 , � 0.6

� 0.4 � c::: 0.2 '0 � 0 � 1 -0.2 o iii -0.4 .. ; ·0.6

-0.8

.,o;- -----;';o,o ---;;':2oO------;3�0 ---;;,40C-----;5� 0 ------c6;!;;-0---;;':7 0;------;8Oc-0 --;0;90---;;' 00 time(sec)----->

FIG. 4: VARIATION OF ROTOR ANGLE WITH TIME

0.5,---�-�-�-,___�-�-�-,___�-____,

0.4

1 0.3

� 0.2

� 0.1 � � 0 .� g -0.1 � io -0.2 � -i -0.3

-0.4

-0.50 10 20 30 40 50 60 70 80 90 100 time(sec}------->

FIG. 5: V ARIA nON OF ROTOR ANGULAR VELOCITY WITH TIME

FIG. 6: PHASE PORTRAIT OF QI-pl

FIG. 7: PHASE PORTRAIT OF QI-pl

VI. CONCLUSIONS

This paper presents a theoretical framework for designing block back stepping controller for nonlinear underactuated TORA system. The proposed control algorithm ensures the global asymptotic stability of the origin of the TORA system. At the outset of the design, a change of coordinates has been introduced to transform the state model of the

underactuated system into a block strict feedback form, which is convenient for backstepping design for Multiple Input Multiple Output systems. Then a nonlinear block­backstepping control law has been designed for the TORA system.

The proposed algorithm is more flexible and efficient than other backstepping-based control algorithms in a sense that it requires fewer numbers of theoretical assumptions to be satisfied for its successful applications. On the positive side, it preserves all the salient advantages of backstepping based control design for nonlinear systems. Moreover, the proposed control law is capable of addressing stabilization issue for a large number of underactuated systems without significant modifications.

The effectiveness of the proposed control algorithm has been established in simulation studies. The experimental results establish the fact that the speculative design of the control law is apt for real-time applications. Finally, to conclude, the authors take liberty to claim that the proposed control law is not only able to ensure theoretical asymptotic semi global stability for a large class of underactuated systems, but it also exhibits an excellent performance in the case of practical applications.

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