LQR Based Type-2 Fuzzy Controller For Double Inverted Pendulum

LQR Based Type-2 Fuzzy Controller For Double Inverted Pendulum

Gaurav Malik*, M. J. Nigam

E & CE Department IIT Roorkee Roorkee, India

Abstract

The inverted pendulum system, which has been focused by control engineering, is among the most important and classical problems. While designing a fuzzy controller for such system, two main problems are controlled namely rule number explosion and uncertainty in the system. As remedy to these problems, a fusion function based on LQR control is applied before fuzzy controller that reduces number of inputs. The second problem uncertainty is taken care by type-2 fuzzy controller, as the degree of fuzziness is greater than conventional fuzzy controller. Therefore, both the schemes are combined to make the controller more efficient and robust. MATLAB simulation and comparison shows that the control effect is perfect.

Keywords- Fusion Function, Fuzzy Control, Double Inverted Pendulum (DIP), LQR Control, IT2FS, Uncertainty

1. Introduction

   The inverted pendulum in control engineering is an important classical problem that reflects many crucial questions during its control process. This system is typically non-linear, complicated, high-ordered and highly uncertain. Here, we are using double inverted pendulum system, which has even more control targets that tests the controller on a higher level.

   The advantage of using fuzzy logic bound system over classical control system is that they tend to have

   smoother control, require little mathematical knowledge of model behavior, noise immunity and uncertainty handling. These use expert knowledge for obtaining results. Due to such characteristics, it has

   become very popular in very short time. It is being used in vast area of research and applications that shows its versatility and power.

   In general, we can say that uncertainty is caused by the lack of information i.e. incomplete, fragmentary, partially reliable, imprecise, vague and sometimes contradictory nature of information. Fuzzy reasoning allows us to handle this uncertainty. So, using type -1 fuzzy sets is a sensible option to use, when uncertainty is involved (Zadeh, 1975 [1]). However, using accurate membership function for something uncertain is not reasonable. So, to handle such uncertainties, type-2 fuzzy sets are used (Mendel, 2001 [2]). As we know that uncertainties cannot be separated from real systems, the research of novel methods to handle incomplete or less reliable information is of great interest (Mendel, 2001 [3]). This makes type-2 fuzzy logic controller as a good sensible choice.

   It is not a trivial task to apply fuzzy control strategy to large-scale complex system. These require different and special approaches for modeling and control. An exponential increase in the number of control rules is observed with the increase in number of inputs. If we assume that we have t input variables and we have defined d fuzzy sets for each variable, then the total number of rules reaches to dt. This problem is referred as rule explosion problem.

   Now, high number of controller input dimensions and excessive inference rule, reduces inference speed and correctness. So it becomes difficult to design and degrade controllers performance. The reduction of fuzzy controllers dimensions and number of fuzzy inference rules is of great research interest. To tackle this problem, Raju and Zhou [4, 5] used the idea of hierarchical structure in designing a fuzzy system, in

   which the input variables are given to low dimensional fuzzy logic units (FLUs) and their outputs are fed as input variables to fuzzy logic units (FLUs) in next layer.

   Joo [6] proposed a method where he considered the fuzzy rules as fuzzy rule vectors to convert the given multi input fuzzy system to two layered hierarchical fuzzy systems. In [7-10], using genetic algorithms, corr. parameters for the sensory fusion function method are found automatically. Fusion function can be achieved by many ways. In [11, 12], LQR gains are used to reduce the fuzzy controllers rule base and simultaneously importing LQR controllers features in the control action of a double inverted pendulum. Similarly, in [13] this idea was implemented to design a DSP chip based real-time motion control for rotary inverted pendulum system.

   In our work, an attempt for the reduction of inference engine for large scale system is made by using a LQR based fusion function and it is combined with a type 2 fuzzy controller system such that the uncertainties in the model are well handled. Finally, a comparison has been made between our controller and LQR fuzzy controller by Wang and Sheng [12]. Paper is organized as mathematical Formulation of double inverted pendulum system in section 2, theory behind and procedure involved in design of fusion function based on LQR in section 3,basic theory of Interval type-2 fuzzy logic in section 4. In section 5, type-2 fuzzy controller design is described and results found are given with discussion in section 6. Finally, the conclusion and future work is proposed in section 7.

2. Formulation of double inverted pendulum system

   The double inverted pendulum is a device that is composed of a cart on which the pendulum is fixed through a mounted shaft and the cart moves along a guided rail [14]. To measure the cart position, a photoelectric coder is installed at one end of the rail. Rotor shafts connect cart to the pendulum as well as both pendulums to each other and photoelectric coders at the connections measure the angles of upper and lower pendulum. The pendulums can move right and left on horizontal guide rail around respective motor shaft thus to make the inverted pendulum

   stable at the vertical position. The schematic diagram of DIP and the notations used with their values are given below.

   M (mp1, mp2 , mj )	Carts mass (that includes first pole, second pole, joint) 5.8 kg (1.5kg, 0.5 kg, 0.75kg)
   p1, p 2	The angle made by pole 1(2) and vertical direction (rad)
   Lp1 (lp1 ), Lp2 (lp2 )	Length of pendulum first (2 lp1 ) and length of second pendulum (2 lp 2 ), 1m, 1.5m
   G	Centre of gravity 9.8 m/s2
   F	Force applied to cart

   M (mp1, mp2 , mj )	Carts mass (that includes first pole, second pole, joint) 5.8 kg (1.5kg, 0.5 kg, 0.75kg)
   p1, p 2	The angle made by pole 1(2) and vertical direction (rad)
   Lp1 (lp1 ), Lp2 (lp2 )	Length of pendulum first (2 lp1 ) and length of second pendulum (2 lp 2 ), 1m, 1.5m
   G	Centre of gravity 9.8 m/s2
   F	Force applied to cart

   Figure 1: DIP schematic diagram Table1 : DIP Parameters

   Lagrange equations are used to derive the equations of motion of the above system

   i

   i

   d dL dL Q dt dqi dqi

   Where L = T V is a Lagrangian, Q is a generalized force vector that acts in generalized coordinates qs direction. It is not taken in account for the formulation of kinetic energy T and potential energy

   V. Kinetic and potential energies of the system are given by the sum of all the energies of carts and pendulums.

   T 1 (m m m m )x2 ( 2 m l 2 2m l 2

   Inverted Pendulums are usually following

   2 c p1 p 2 j

   3 p1 1

   p 2 1

   pproximations are used:

   2m l 2 )2 1 m l 2 2

   p3 1 p1 6 p 2 p 2 p 2

   • (mp1lp1

   • 2mp 2lp1

     cos( p1 p 2 ) 1, sin( p1 p 2 ), cos( p1 ) cos( p 2 ) 1,

     sin( p1 ) p1, sin( p 2 ) p 2

     2mp3lp1 )xp1 cos p1 mp 2lp 2 xp 2 cos p 2

     Linearization is made at balance position; so we get

     2m l l

     cos(

     )

     the LTI state space model [] as:

     p 2 p1 p 2

     p1 p 2

     p1 p 2

     x

     0 0 0 0 0 0 x 0

     V mp1glp1 cos p1 2mp3 glp1 cos p1

     p1

     0 0 0 0 0 0

     p1 0

     0 0 0 0 0 0

     p 2 0

     m g(2l

     cos

     • l cos )

       p 2 u(t)

       p 2 p1

       p1 p 2 p 2

       x 0 0 0 0 0 0 x 1

       Using the equations above the Lagrangian L of the

       0 14.2545

       4.0090 0 0 0

       1.1818

       p1

       p1

       system is given as

       0

       14.2545 21.1077 0 0 0

       0.1818

       L 1 (m m m m )x2 ( 2 m l 2 2m l 2

       p 2

       p 2

       x

       2 c p1 p 2 j

       3 p1 1

       p 2 1

       0

       2m l 2 )2 1 m l 2 2

   • (m l

   • 2m l

   p1 0

   p3 1

   p1 6

   p 2 p 2 p 2

   p1 p1

   p 2 p1

   y(t) diag.1 1 1 1 1 1

   p 2 0

   u(t)

   2mp3lp1 )xp1 cos p1 mp 2lp 2 xp 2 cos p 2

   x

   0

   2m l l cos( ) m gl cos

   0

   p 2 p1 p 2

   p1 p 2

   p1 p 2

   p1 p1 p1

   p1

   2mp3 glp1 cos p1 mp 2 g(2lp1 cos p1 lp 2 cos p 2 )

   0

   p 2

   Now the Lagrangian is differentiated by and

   to yield Lagrange equation as given below

   Throughout the paper this linearised model of double inverted pendulum is used.

   d dL dt dp1

   • dL 0

     d p1

3. Design of fusion function based on LQR

   d dL dL

   Since we are going to use LQR control technique

   or explicitly:

   dt dp 2

   0

   d p 2

   features in our controller which is applicable to linear state space model, so we define the state-space equations for our system.

   ( 4 m l 2 4m l 2 2m l 2 )

   • (m l

   • 2m l

   X (t) AX (t) Bu(t)

   3 p1 p1

   p 2 p1

   j p1 p1

   p1 p1

   p 2 p1

   X (t) CX (t) Du(t)

   2mjlp1 )xcos p1 2mp 2lp1lp 2p 2 cos( p1 p 2 )

   Performance index J is chosen as

   2m l l

   2 sin(

   ) (m l

   2m l 1

   p 2 p1 p 2 p 2

   p1 p 2

   p1 p1

   p 2 p1

   J =

   ( XT (t)QX (t) uT (t)Ru(t))dt

   2mjlp1 )g sin p1 0 2 0

   m l xcos

   2m l l

   cos(

   )

   where Q and R are chosen to be positive semi-

   p 2 p 2

   1 m l 2

   p 2 p 2 p1 p 2 p1

   • 2m l l 2 sin(

   p1 p 2

   )

   definite matrices. They determine the matrix K of the

   optimal control vector

   3 p 2 p 2 p 2

   p 2 p1 p 2 p1

   p1 p 2

   u(t) KX (t)

   mp 2lp 2 g sin p 2 0

   A more compact matrix form of the Lagrange equation for the DICP system is given below D( ) C( ,) G( ) Hu

   The stationary point of the system is given by

   (x,p1,p2 , x,p1,p2 , x) (0,0,0,0,0,0,0).

   Now introducing a small deviation around the stationary point and expanding it using Taylor series; also, in the stable control process of the Double

   Minimizing the performance index J, the elements of matrix K are found. Then

   u(t) KX (t) R1BT PX (t)

   is optimal for any initial X(0) state. Where P is the solution of algebraic Riccati equation (given below) and K is the linear optimal feedback matrix. AT P PA PBR1BT P Q 0

   Here, we have chosen as

   Q diag([10 60 80 0 0 0]) and R 1

   After solving, we get

   K [10 275.2453 515.650216.2044 22.1046 111.9285]

   Now, using this K matrix, a fusion function F(X) is constructed as suggested by Wang and Zheng [11]. It is described as

   in the problems of real world. Zadeh proposed these sets in 1975, they are fuzzy-fuzzy sets in which the

   1 Kp1

   Kp 2

   Kpn

   0 0 0

   F ( X )

   0 0 0

   K K K

   K

   p1

   p1

   p 2

   p 2

   pn

   pn

   n n

   Where,

   K (K

   i1

   )2 (K )2

   pi pj

   pi pj

   j1

   After solving, it comes out to be

   F( X ) 0.01939 0.5338 1 0 0 0

   0 0 0 0.14477 0.19749 1

   Then, input variables dimensions are reduced. The error (E) and error change (EC) may be obtained by F ( X ) as :

   EC

   EC

   E F ( X ) X T

   Figure 2: Rule base reduction methods compared with d 5.

   membership grades are itself type-1 fuzzy sets. These sets express the degree of uncertainty and non- determinist truth with which an element belongs to the whole set.

   Since, our fusion function reduces the number of

   inputs for the fuzzy controller to two, namely, error

   If f (u) 1, u J u , J u [0,1]

   p p p

   p p p

   (E) and error change (EC). We can formulate the number of rules i.e. d2. The Table (2) below shows

   An interval type-2 fuzzy set (IT2FS) A can be

   1

   1

   characterized as:

   the comparison of different rule reduction methods. Table 2: Comparison of different reduction methods

   A

   uJ [0,1]

   pP p

   (x,u)

   pP

   [uJ [0,1] u

   p

   p

   ]

   ]

   1

   1

   x

   Methods used to reduce the no. of rules	The t 1 no. of variables
   	Even	Odd
   Sensory fusion	d t /2	d (t 1)/ 2
   Hierarchical	(t 1).d 2
   Combinational	((t / 2) 1).d 2	(((t 1) / 2) 1).d
   LQR-fusion	d 2

   Methods used to reduce the no. of rules	The t 1 no. of variables
   	Even	Odd
   Sensory fusion	d t /2	d (t 1)/ 2
   Hierarchical	(t 1).d 2
   Combinational	((t / 2) 1).d 2	(((t 1) / 2) 1).d
   LQR-fusion	d 2

   Where the primary variable p, has domain P; the

   secondary variable u U ; has domain J at each

   p

   p P ; J is called the primary membership of p

   p

   and the secondary grades of A

   are all equal to 1.

   Clearly, means

   A : P {[u,v]: 0 u v 1}.

4. Interval type-2 fuzzy logic

The usage of type-2 fuzzy sets provides us the advantage of handling the uncertainty and inaccuracy

Union of all the primary membership conveys uncertainty about A , which is also known as footprint of uncertainty (FOU) of A . The shaded area (Mendal, 2000) which is bounded by an upper and lower membership function as shown in figure below:

FOU( A ) = pP Jp {( p,u) : p Jp [0,1]}

Here, the upper membership function (UMF) and lower membership function (LMF) of A are type-1

membership function as shown in Figure 3.

B ( y)

M