Deadbeat Control of Linear and Non Linear System using Signal

MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

Deadbeat Control of Linear and Non Linear System using Signal Correction Technique Soumen Paul, Haldia Institute of Technology, India, [email protected] Asim Halder, Haldia Institute of Technology, India, [email protected] Asoke Kumar Nath, Jadavpur University, India, [email protected] Abstract- This work presents the deadbeat control technique to mitigate transient oscillations of both linear time invariant (LTI) and nonlinear continuous system based on Signal Correction Technique (SCT). In SCT a suitable corrected signal is generated and incorporated along with a reference input to the system through the feedback loop. Here, the generalized state space model is considered to derive the SCT based deadbeat control law of the system response. The SCT based deadbeat representation with state space equations is demonstrated with some examples of higher order systems with the reference input as step, ramp and any linear type input. For the deadbeat realizations, appropriate strictly increasing polynomial curves are considered to find out the rise time of the system response. Then the output response is made same as the reference input to the system from rising point onwards. Furthermore, the stability analysis of the given systems has been demonstrated using Lyapunov Stability theory. In case of nonlinear system, the system dynamics of the Inverted Pendulum is selected in the aspect of nonlinear control theory, with an emphasis on feedback linearization. This technique may be very much useful in realizing the transient performance of a system without incorporation of any controller, restriction of system parameters and experimental data. Keywords: Signal Correction Technique, Deadbeat, Feedback Linearization, Lyapunov Stability theory

I.

INTRODUCTION

A system is broadly classified as linear and nonlinear depending on the nature of differential equation model. If the principle of homogeneity together with the principle of superposition holds good for a certain range of inputs the system is linear in that range of inputs. The systems which are not linear are referred to as nonlinear. There are various other types of classifications of dynamic systems [1] like continuous and discrete, stationary and time varying, deterministic and nondeterministic systems for convenience of their representation and analysis. Of all these systems, the analyses of linear deterministic continuous systems are the simplest. Unfortunately, most control systems we encounter in our everyday life are nonlinear in nature. So, in order to derive the benefits of mathematical advantage of linear system analysis, the nonlinear systems are approximated by linear models, whenever permissible, by suitable techniques known as linearization. Most process control systems [2,3,4,5,6] are inherently nonlinear, The results described in [4] shows the applicability and accuracy of the proposed designs in linear Mechatronics motion control applications for achieving desired deadbeat specification. The concept of deadbeat control is very old in control engineering. The behavior of the output of a control system in the transient state as well as in the steady state following the application of an input or sudden change of set point is a very important performance measure of any system whether linear or nonlinear. In many applications, the overshoot and undershoot or ringing behavior in the output cannot be tolerated. In such a system one prefers what is known as dead beat response of the output. The deadbeat response has the following desirable characteristics: i) Zero steady-state error. ii) Minimum rise time. iii) Minimum settling time. iv) Less than 2% overshoot/undershoot. v) Very high control signal output. In the past many approaches were taken for realizing deadbeat control. A new approach, called signal correction technique (SCT) for the deadbeat realization is being researched by many workers. In the

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MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

publications including the recent ones, show that though the signal correction technique has many advantages, not much work of SCT designed deadbeat control been reported so far. These led the authors to be interested in the application of SCT in the deadbeat realization of linear and nonlinear systems. Signal Correction Technique (SCT) is a technique of injecting a suitable additional signal to the system along with the reference input to get deadbeat response. This technique has also been used for removal of system instability and modification of the system nonlinearity. In this work, SCT is used for deadbeat realization of linear systems and non linear systems. The required suitable signal may be generated by using states of the system. Fig. 1 represents the corrected signal in control system.

Corrected Signal

State Input

+

Output Plant

+

+

Feedback path

Closed loop system Figure 1. Signal Correction Technique

Various techniques [7,8,9,10,11,12,13,14] have been suggested for realizing deadbeat transient response of linear control systems. In a few suggested methods, a deadbeat controller is designed and put in the forward path of the control loop. The integration between robust Quantitative Feedback Theory-based controller (QFT) and self tuning deadbeat algorithm [10] has improved the grain dryer control system‘s performance. The work [11] proposes a new design methodology for ripple-free deadbeat control of nonlinear systems in discrete-time with the approach of two steps using state feedback linearization and then applying the Diophantine equations design approach. Our work presents the techniques of deadbeat transient performance of higher order linear system based on SCT scheme using pulse signal to the linear continuous system. Some works [15,16,17,18,19,20,21] other than SCT had been done for deadbeat control of discrete time linear control system with the representation of state space equations. It is observed in some applications, such as biological control systems, it may not always be possible to incorporate a controller inside the system. Numerous research papers had been published on deadbeat control of discrete time linear control system represented in the state variable form. Design of deadbeat controller with limited output [15] is studied with the method based on the introduction of an additional polynomial into the transfer function of the controller. and it was found that the quality of the deadbeat control system is influenced by the accuracy of the transfer function of the object. In [16], deadbeat control is extended to linear multivariable discrete-time generalized state-space systems using algebraic methods. Modification of digital controller algorithms is achieved in [17] using phase variables as state variables. The existence and construction of deadbeat control laws in discrete time using the state-space techniques is described in [18, 19], a general minimum-time state deadbeat controller is presented

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for a class of simple Hammerstein systems. Dahlin‘s control algorithm [20] is presented for improved identification and control of discrete time nonlinear system. The authors [21] developed a theoretical framework to analyze the trade-off between the communication cost and the deadbeat control performance using Markov chain model In section II, the background of this research work is described. Problem formulation of a deadbeat is explained in section III. The stability analysis using Lyapunov‘s Stability theory is elaborated in section IV. Section V demonstrates the results and figures of deadbeat for some examples with step, ramp and linear type input. Deadbeat representation in nonlinear system is done in section VI. Conclusion and future scope of the work is described in section VII. II.

BACKGROUND AND MOTIVATION

The signal correction technique (SCT) is a method where a suitable signal is generated by an algorithm using the states of the system and added with the command signal to implement the deadbeat response. No controller has to be incorporated in the control loop nor is any signal-shaping required. In references [5, 22], a general formulation for SCT for dead beat response of linear systems of any order has been suggested, but the algorithm has been implemented only with some parameter restrictions. It has been indicated that such a signal would be difficult to find by adopting the conventional continuous data control techniques. They suggested a computer simulation technique to resolve this problem and illustrated with a lightly damped linear second order system. The desired response is chosen such that after switching, the system response follows a straight line trajectory in the phase plane. The nature of the added signal, they obtained, as a single pulse of duration T, the sampling period of the digital simulation, applicable only at the switching instant. The authors have not suggested any method for finding the correct (desired) value of T. The implementation of the dead beat response for a system with any given ξ and ωn would require unending search for T. Thus, the suggested method has implementation problem. Based on experimental results, they have provided a plot of ξ vs. T and ξ vs. amplitude of the pulse. For a given system, knowing ξ, the values of T and the amplitude of the pulse can be obtained using these plots. But, they have indicated that the product of ωn and T should be 10.425. This shows that these plots and the cited relation will help only in implementing the dead beat response of systems with specific combinations of ξ and ωn. That restriction of specific combination has been eliminated in this work. During the last few decades, there has been extensive research on RF-DB (Ripple free Deadbeat) control systems [7, 9, 11] and [23,24,25] and various schemes have been proposed, aiming at the application of such modern techniques to the control of widely diverse plants. In [26] the authors investigated the best closed loop performance that is achievable by structured static state feedback controllers based on limited model information. The present work suggests the approach based on the SCT concept that does not require any restrictions on the system parameters. This corrected signal can be generated through the proper simulation by using the relation between state space variables within the system and the desired output. III.

FORMULATION OF DEADBEAT

In this work multiple numbers of deadbeat transient responses y (t ) may be considered with different rising time. y (t ) is the desired output and a polynomial type of function in time domain.

Y (s) : output signal

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E (s) : actuating signal U (s) : reference input G(s) 

Y (s)

 forward path transfer function

E (s)

H ( s) : transfer function of the feedback elements.Here H(s)  1 as no transfer function in feedback a n  1s n1  a n2 s n2  an3 s n3  ............  a 2 s 2  a1s  a0 Y ( s)  n2  .b n3  .............  b s 2  b s  b (1) E ( s) s n  b s n1  b 2 1 0 n1 n 2 s n3 s Let us consider the open loop transfer function G(s) of an nth order linear continuous system in the form: If the G(s) 

loop is closed around the forward path transfer function G(s) with unity feedback, we have E(s) = U(s)-Y(s). Therefore, the closed loop transfer function Y(s) will be given by U(s) Y (s)



U (s)



G(s)

(2)

1  G(s)

a n  1s n1  a n2 s n2  a n3 s n3  ...........  a1s  a0 s n  a n  1  bn1 s n1  a n2  bn2 s n2  .............  a1  b1 s  a0  b0 



Let, X(s) 

Y ( s) 



(3)

1 U ( s) n n  1 s  a n  1  bn1 s  a n2  bn2 s n2  .............  a1  b1 s  a0  b0 





s n1  a n2 s n2  a n3 s n3  ...........  a1s  a0 U ( s) s n  a n  1  bn1 s n1  a n2  bn2 s n2  .............  a1  b1 s  a0  b0 









So, Y(s) is given by, Y (s)  an  1s n1  an2 s n2  an3s n3  ...........  a1s  a0 X (s)

(4)

(5)

(6)

Hence Y(s) can be written as  n 1  or Y ( s )    ai s i  X ( s )   i  0 

(7)

From (4), (5), (6), (7) with choosing the state x = x1 as the first state and its derivatives as other states it is reduced to a set of n first-order differential equations given below.

x1  x 2 x 2  x3

x3  x 4 .

(8)

. . x n2  x n1 x n1  x n









x n  a0  b0 x1  a1  b1 x 2  a 2  b2 x3  .......  a n2  bn2 x n1  a n  1  bn1 x n  u (t ) n 1 or x n  u (t )   a i bi xi  1 i0





(9) (10)

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MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

From (5), the output response y(t) in time domain y (t )  a0 x1  a1 x 2  a 2 x3  ...........................  a n  2 x n  1  x n (11) n2 or y (t )  x n   ai xi  1 i0 (12) Thus, considering phase variables as the state variables, the dynamics of the system can be readily written in the

general state variable in form of (13) and (14)

X  AX  BU

(13)

Y  CX  DU where X is the state vector, U and Y are the input and output vectors respectively.

(14)

For a single input single output (SISO) system, u(t), and y(t) are scalars. Further, if u(t) is unit step and y(t) is not deadbeat, then have to add another corrected signal f ( x1 , x 2 , x3 ,...........x n) through another feedback loop to ensure deadbeat response. Thus, for all t  t 0 , y(t) will not have any overshoot or undershoot and y(t) will be equal to u(t), where t 0 is the time when y(t) attains the steady state condition for the first time. For the time,

0  t  t 0 y(t) is strictly increasing. Thus y (t ) will give the deadbeat response if and only if the following two conditions hold: y(t) is strictly increasing for 0  t  t 0 y(t )  u(t ) for t  t 0

i ii

i.e. y (t )  0 for 0  t  t 0

(15) (16)

Then the state and output equation of the deadbeat system can be written as follows. As f ( x1 , x 2 , x3 ,...........x n , t ) is added to the system alongwith reference input u(t) for the deadbeat, then using(10)









x n  a0  b0 x1  a1  b1 x 2  a 2  b2 x3  .......  a n2  bn2 x n1  a n  1  bn1 x n  u (t )  f ( x1 , x 2 , x3 ,...........x n , t )

(17)

n 1 or x n  u (t )  f ( x1 , x 2 , x3 ,...........x n , t )   a i bi xi  1 i0 n 1 or x n  u1 t    a i bi xi  1 i0 where u1 (t )  u1  u(t )  f ( x1, x2 , x3 ,...........xn , t )









(18)

Now

x  Ax  Bu 1

y  Cx  Du1

(19)

(20)

 x1    x2  x   3 x  . , (21) nx1   .  .     x n 

B nx1

where D=null matrix

5

0  0    0     .  .    0  1   

(22)





c1xn  a0 a1 a 2  a n  2 1

(23)

MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

 0   0  0          Anxn               a  b   0 0 

1 0 0  

0 1 0

                  an  2  bn  2   an  1  bn 1   

0 0 1

0 0 0



 a2  b2   an 3  bn 3 

 a1  b1

0 0 0

(24)



u1  u1(t)  u(t)  f(x1, x 2 ,.x3.........,x n )

(25)

Now the objective is to find f(x1, x 2 . . . . , x n ) so that conditions (15) and (16) are satisfied. y(t )  a 0 x1  a1x 2  a 2 x 3 ...........  a n 2 x n 1  x n

(26)

Taking the derivative with respect to time, we have  (t)  a 0 x 1  a1x 2  a 2 x 3 ...........  a n2 x n 1  x n y

(27)

Substituting the values from the state space equations (8), (9), (10) and new input u1(t) in (25), we get the firstorder differential equation given below.

y (t)  a 0 x 2  a1x 3  a 2 x 4  ......... a n2 x n



 















 a 0  b 0 x1  a1  b1 x 2  a 2  b 2 x 3  a 3  b3 x 4  ................... a  b n1 x n  n 1 u(t)  f(x1, x 2 , x 3 ,...........x n )

(28)

f(x1 , x 2 , x 3 ,...........x n )  y (t ) - a 0 x 2  a1x 3  a 2 x 4  ......  a n 2 x n  a0  b0 x1  a1  b1 x 2  a 2  b2 x3  a3  b3 x 4  .. . .  a n  1  bn1 x n  u (t )

or



IV.



(29)

STABILITY OF A LINEAR SYSTEM WITH LYAPUNOV STABILITY THEORY

Stability analysis is an important factor in controller design. Therefore, the stability calculation is necessary for the system with SCT of deadbeat control. In this context, Lyapunov stability method is used to check the stability when SCT is incorporated with the system. Lyapunov stability theorem (Ogata, 2002) is one of the useful methods for the stability analysis of nonlinear as well as linear systems. In general a Linear Time Invariant (LTI) system is considered as-

x  Ax ;

where, x  R n and A is an n  n matrix

(30)

According to Lyapunov theorem, the Lyapunov function in quadratic form can be written as(31)

V  x T Px

where, matrix P  R nn is symmetric. The derivative of (30) with respect to the vector field x  Ax is:

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MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

V  x T Px  x T Px

 x T AT Px  x T PAx

  x T Qx T where, A P  PA  Q

(32)

From (32) it can be said that Q is a symmetric matrix. NowIf P and Q are both positive definite, then according to Lyapunov 1st Theorem the origin of the system

i.

(30) is asymptotically stable ii.

If Q is positive definite and P is negative definite or indefinite, then in both cases Lyapunov function

‗V‘ can take negative values in the neighborhood of the origin, so according to Lyapunov 3rd Theorem the system (30) is unstable. Therefore, it can be said that the matrix A  R nn is stable if and only if for any given positive definite symmetric matrix Q, there exist a positive definite symmetric matrix P which satisfies Lyapunov matrix (32). Moreover, if A  R nn matrix is stable then the solution P can be written as This is said to be the unique solution of Lyapunov matrix equation (32).

 T P   e tA Qe tA dt 0

(33)

Therefore by choosing suitable values of a n and bn (where n= 0, 1, 2.......... n) the positive definite symmetric matrix P  Rnn can be estimated for the stability of the given system.

V.

RESULTS AND FIGURES WITH EXAMPLES

Let us consider the example of following third order system.



Let the open loop transfer function i.e. G(s)

The closed loop of transfer function is

X s  

s 2  a1s  a0 s 3  b2 s 2  b1s  b0

s 2  a1s  a0 Y s   U s  s 3  1  b2 s 2  a1  b1 s  a0  b0 

U s  3 2 s  1  b2 s  a1  b1 s  a0  b0 



(34)

(35)

(36)



Y s   s 2  a1s  a0 X s 

(37)

This system may be represented in state space by the following equations: .

x  Ax  Bu

(38)

y  Cx  Du

(39)

Where x is the state vector, u and y are the input and output vectors respectively. x1  x 2

(40)

x 2  x3 x3  a0  b0 x1  a1  b1 x 2  a 2  b2 x3  u t  y t  a 0 x1  a1 x 2  x3



(41)



(42) (43)

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MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

A. Examples with Step and Ramp Input Let us take the two following transfer functions of 3 rd order and 4th order.

Example V.I:

G(s) 

s 2  2s  30 s 3  7s 2  6s  10

Example V.II:

G(s) 

s 3  3s 2  2s  40 s 4  10s 3  40s 2  20s  5



If u(t) is the unit step and y(t) is not deadbeat, a corrected signal f x1 , x 2 , x 3

 will be added through a

feedback loop making y(t) a deadbeat response i.e. for all t>t0, y(t) will not have any overshoot or undershoot and y(t) will be equal to u(t), where t 0 is the time when y(t) attains the steady state condition for the first time. For the time 0  t  t 0 , y (t) is strictly increasing function of t. From (15) and (16) y (t ) will give the deadbeat response if and only if the following two conditions hold. i. y(t)is strictly increasing 0  t  t 0 i.e. y (t )  0 for 0  t  t 0

(44)

ii. y(t )  u(t ) for t  t 0

(45)

Then the state and output equation of the deadbeat system are given by

 0  x1    x2    0  x    3  a0  b0

1 0



0 1  a 2  b2

   x1  0        x 2   0   x3  u t   f x1 , x 2 , x 3 , t  

(46)

   a1  b1   Now the objective is to find f  x , x , x  so that (44) and (45) are satisfied. 1

2

3

y (t )  a0 x1  a1x2  x3 or y (t )  a0 x1  a1x 2  x3

(47)





or y (t )  a0 x2  a1x3  a0  b0 x1  a1  b1 x2  a 2  b2 x3  u (t )  f ( x1, x2 , x3 , t ) or f x1, x2 , x3 , t  y (t )  a0  b0 x1  a1  b1  a0 x2  a 2  b2  a1 x3  u (t )



y (t )











is the deadbeat transient response after the additional or corrected signal

and this

(48) (49) (50)

f x1 , x2 , x3  into the system

y (t ) must hold the above conditions (44) and (45). In this technique there is no restriction of rise time

of the system output for steady state condition. Hence some arbitrary parabolic equation or linear equation y(t), which is strictly increasing for tt0 where t 0 is the rise time

of the output response y(t). The y (t ) will give the dead beat response if and only if the following conditions hold.

y(t) is strictly increasing for 0  t  t 0 i.e. y ' (t )  0 for all 0  t  t 0 ii y(t)  u(t )  t for all t  t 0 i

(70) (71)

Let us consider y(t)= Y1(t)=0.3792.t + 0.3t2. Since unit ramp input is given, The polynomial Y1(t) intersects vertically. i.e. Y1(t)=t (as unit ramp input is given) at the point t=2.0693. So the rise time of output transient response is t0=2.0693. Hence the output y(t)=t is the deadbeat realization with steady state condition at t>2,0693. In the above example V.I, the transient output with unit ramp input is given in Fig. 13. Here y(t)= Y1(t)= 0.3792.t + 0.3t2. is considered as the polynomial curve for the deadbeat response y(t) in the interval 0  t  t 0 .

Hence y(t)  Y 1t   0.3792.t + 0.3t 2 , for 0  t  t0

y(t )  u(t )  t

(72)

for t  t 0

(73)

It means y(t) reaches the steady state condition of the system at time t 0=2.0693. The deadbeat response and

corrected

y (t )

f x1 , x2 , x3  signal given in Fig.15 and Fig. 16 respectively.

Now y(t)  0.3792  0.6t

y (t) is strictly increasing in 0  t  t0 ,where t0=2.0693, where 0.3792 is added in the simulation block of step1 and

t is given as a ramp input in Fig.14. In this simulation block of step 3, step time is given as 2.0693. The

initial value of step input is zero and final value of step input is 1. These are adjusted in the MATLAB simulink



to ensure the additional or corrected signal f x , x , x 1 2 3

 in the feedback loop holding the above two

conditions. In the example V.I

a0  b0  =40, a1  b1  a0  =-22 and a2  b2  a1  =6. The above three values are imposed in the simulation block of G1, G8 and G7 respectively in the Fig. 14. If y(t )  Y 2 t   0.3.t + 0.7t.2, for 0  t  t0 then the rise time of the dead beat system will improve. In this case rise time t0 is 1.0. Thus the deadbeat output of the system y(t) reaches to steady sate condition at t0 =1.0. Hence Y (t )  Y 2 t   0.3.t + 0.7t2, t0 is 1.0, i.e

and y(t )  t

y(t )  Y 1t  for

0  t  t0

for t  t 0

Thus, y(t) reaches the steady state condition of the dead beat system at time t 0=1.0. Just like previous cases for unit step input even the rise time of deadbeat system improves but the additional or corrected input given to the system is more oscillatory in nature with the high amplitude. The dead beat response of the example V.I with

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MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

y(t)= 0.3.t + 0.7t2 and the corrected signal f x1 , x2 , x3 to the feedback loop are given in Fig. 17 and Fig. 18 respectively. The polynomial curve Y1(t) and Y2(t) are presented in Fig. 19 and Fig. 20 respectively. 5

4

c(t)

3

2

1

0

0

2

4

6

8

10

t in secs

Figure 13. The output response of example V.I with ramp input Corrected Signal Reference Input

Deadbeat Output

Figure 14. The simulation diagram with ramp input for example V.I 10

4

8

3

Corrected signal

y(t)

6 4 2 0 -2

2

4

6

8

1 0 -1

Dead beat response unit ramp input 0

2

-2

10

t in secs

0

2

4

6

8

t in secs

Figure 15. The deadbeat response of example V.I for t0 =2.0693

Figure 16. The corrected signal of example V.I for t0 =2.0693

14

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MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

4 10

3 corrected signal

8

y(t)

6 4 2 0 -2

2

4

6

8

1 0 -1

Dead beat response Unit ramp input 0

2

-2

10

0

2

4

t in secs

6

8

10

t in secs

Figure 17. The deadbeat response of example V.I for t0 =1.0

Figure 18. The corrected signal of example V.I for t0 =1.0

Y1 (t)

Y2 (t)

Figure 19. The polynomial curve Y1(t) considered of example V.I

Figure 20. The polynomial curve Y2(t) considered of example V.I

If u(t) is the linear type input i.e. u(t)= a 0  a 1 t and y(t) is not dead beat, then the corrected signal input f(x1,x2,x3,x4,….,xn) through a feedback loop making y(t) a deadbeat response then y(t)=u(t)= a0  a1t ,

for t  t 0 , where

t0

is the time, when y(t) attains the steady state condition for the first time. For the time,

0  t  t 0 , y(t) will not have any overshoot and strictly increasing function of time t.

The y (t ) will give the dead beat response if and only if the following conditions hold.

i. y(t)is strictlyincreasing for 0  t  t 0 , i.e. y'(t)  0

(74)

ii. y(t)  u(t)  a

(75)

0

 a t for t  t 0 1

For simplicity, let us consider a  1, a  1 For the above example V.I let us consider the polynomial curve, 0 1 y(t)= Y1(t)= 0.3792.t + 0.3t2. Since linear input is considered with a

0

 1, a

1

 1 , the output at steady state

condition is also considered as unity Y(t)=1+t. The polynomial Y1(t) intersects the linear equation u(t)=t+1 at t=3.1332. So the rise time of output transient response is t0 =3.1332.

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MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

Hence

i. y(t)  Y 1t   0.3792.t + 0.3t 2 for 0  t  t0 where t0 is 3.1332

(76)

ii. y (t )  u (t )  t  1, for t  t 0

(77) (78)

Now y(t)  0.3792  0.6t

Here y(t) is strictly increasing function for for 0  t  t 0 . Thus y(t) reaches the steady state condition of the system at time t0=3.1332. The deadbeat response

y (t )

and corrected signal f  x1 , x 2 , x3  is given in Fig. 21

and Fig. 22.

Figure 21. The deadbeat response of example V.I for t0=3.1332

Figure 22. The corrected signal of example V.I for t0=3.1332

If Y (t )  Y 2 t   0.3.t + 0.7t. , then the rise time of the dead beat system will improve. In this case rise t 0 is 2

1.7956. Thus at t0 =1.7956 the output y(t) reaches to steady sate condition.

Hence

y(t )  Y 2 t   0.3.t + 0.7t 2 for 0  t  t0 , t0  1.7956

y(t )  t  1,

(79)

for t t 0

(80)

It means y(t) reaches the steady state condition of the dead beat system at time t 0=1.7956. In this case even the rise time of dead beat system improves but the additional input or corrected signal given to the system is more oscillatory in nature with the high amplitude. The dead beat response of the example 1 with y(t)= 0.3.t + 0.7t2 and the corrected signal f x1 , x2 , x3 to the feedback loop are given in Fig. 23 and Fig. 24 respectively. Table 1 represents the selected curve and rise time of deadbeat controller with different types of reference input such as step, ramp and linear type for example V.I and example V.II. 4

Corrected signal

3 2 1 0 -1 -2

0

2

4

6

8

10

t in secs

Figure 23. The deadbeat response of example V.I with t0 =1.7956

Figure 24. The corrected signal of example V.I for t0=1.7956

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MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

Table 1: SELECTED POLYNOMIAL CURVE AND RISE TIME OF DEADBEAT CONTROLLER Example

Example V.I

Example V.II

Reference Input

Selected curve for Dead beat

Pattern

Response y(t)

Step

Y1(t)=1.7679t-0.5t2

0.707

Y2(t)=10.25 t - 2.5t2

0.10

2

1.00

Step

Y1 (t )  1.5.t - 0.5t

2

0.707

Y1(t)=0.3792.t + 0.3t2

2.0693

Y2 (t )  1.7679t - 0.5t

Example V.I

Ramp



Example V.I

Linear

Non Linear System of Inverted Pendulum

Step

Rise time t0

2 Y 2 t  0.3.t + 0.7t.

1.0

Y1(t)= 0.3792.t + 0.3t2

3.1332

Y 2 t   0.3.t + 0.7t

1.7956

Y1 (t )  1.5.t - 0.5t

2

1.0

2

Y2 (t )  1.7679t - 0.5t

2

0.707

To check stability In example V.I: values of a0  30 ; b0  10 ; a1  2 ; b1  6 ; b2  7 1 0  0 1 0 0   Therefore matrix A  0 0 1  and positive definite symmetric matrix Q  0 1 0 is considered.   40  8  8 0 0 1  0.5  8.1875  1.7250  Then matrix p can be estimated as P   0.5 8.1875  0.5    8.1875  0.5 41.5 

The matrix P will be positive definite if its Eigen values are positive which are calculated as-

1  0.0631, 2  8.2256 and 3  43.1239 So the system considered in example V.I is asymptotically stable. In example V.II values of a0  40 ; b0  5 ;

a1  2 ;

b1  20 ;

a 2  3 ; b2  20 ;

1 0 0   0 1  0  0 0 1 0  Therefore matrix A   and positive definite symmetric matrix Q    0 0 0 0 0      45  22  43  11 0

b3  10

0 0 0 1 0 0 is 0 1 0  0 0 1

considered.

 0.5  4.8268 0.5   4.6078   0.5 4.8268  0.5  5.6990 Then matrix p can be estimated as P    4.8268  0.5 5.6990  0.5     5.6990  0.5 11.3526   0.5 The matrix P will be positive definite if its Eigen values are positive which are calculated as-

1  0.0440 , 2  1.7430, 3  9.9552 , 4  14.4710 VI.

DEAD BEAT REALIZATION OF NON LINEAR SYSTEM

In this section the deadbeat realization of nonlinear system is implemented after the feedback linearization.

17

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MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

The model of Inverted Pendulum is selected in the aspect of nonlinear control theory, with an emphasis on feedback design. As it is seen, feedback is central to control systems, and techniques from differential geometry and dynamic optimization play leading roles. Feedback is used to stabilize and regulate a system in the presence of disturbances and uncertainty, and the main problem of control engineers is to design feedback controllers. In section VI feedback linearization is applied to make the dynamic system of the inverted pendulum, a globally asymptotically stable. The deadbeat realization technique, discussed in section III is implemented to the linearized model of inverted pendulum. A. Feedback Linearization over the Model of Inverted Pendulum Lot of research works [27,28,29] had been done over feedback linearization over the dynamic system of inverted pendulum [6]. To illustrate some of the benefits of feedback, the model of an inverted pendulum, or a one-link robot manipulator. In Fig. 25, the angle θ is measured from the vertical (θ = 0 corresponds to the vertical equilibrium), and τ is the torque applied by a motor to the reVolumeute joint attaching the pendulum to a frame. The motor is the active component, and the pendulum is controlled by adjusting the motor torque appropriately. Thus the input is u = τ. If the joint angle is measured, then the output is y = θ. If the angular velocity is also measured, then y = (θ, θ˙). The pendulum has length 1 m, mass m kg, and g is acceleration due to gravity. The target is to restore the pendulum bob vertically with the application of motor torque (  ). Fig. 25 represents the inverted pendulum or one-link robot arm. From Fig. 26 it is clear that there are two force (component of force) acting perpendicularly on the bob of the pendulum. I.

Torque due to gravity.

II.

Torque due to generated acceleration in the bob.

Hence,   I  mg sin   l  ml 2  mgl sin 

(81)

Now if l=1, then (81) becomes m  mg sin   

(82)

The model neglects effects such as friction, motor dynamics, etc. We begin by analyzing the stability of the equilibrium θ = 0 of the homogeneous system m  mg sin   0

(83)

corresponding to zero motor torque (no control action). The linearization of (83) at θ = 0 is Since sin θ ≈ θ for small θ, this equation has general solution e t

g

 e

t g

and, because of the first term

(exponential growth), θ = 0 is not a stable equilibrium for (83). This means that if the pendulum is initially setup in the vertical position, then a small disturbance can cause the pendulum to fall. We would like to design a control system to prevent this from happening, i.e. to restore the pendulum to its vertical position in case a disturbance causes it to fall. One could design a stabilizing feedback controller for the linearized system m  mg  

(85)

and use it to control the nonlinear system (82). This will result in a locally asymptotically stable system, which may be satisfactory for small deviations θ from 0. For globally asymptotically stable an approach is adopted, which is called the computed torque method in robotics, or feedback linearization. This method has the potential

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MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

to yield a globally stabilizing controller. The nonlinearity in (82) is not neglected as in the linearization method just mentioned, but rather it is cancelled. This is achieved by applying the feedback control law.

  mg sin    '

Where

(86)

 ' is a new torque input, and results in the closed loop system is (87)

m   '

A linear system described in (87) is a linear system with input  ' . By the idea of feedback linearization of a non linear system, the new input has to be given in such a way that the system is stabilized, i.e. the pendulum is stabilized. Indeed, let‘s set  '   k1  k 2 . , where k1 and k 2 two arbitrarily taken values. Thus (87) becomes m  k1  k 2  0

(88)

If the feedback gains are selected as k1 = 3m, k2 = 2m (for example), then the system (88) has general solution

e t  e 2t , and so θ = 0 is now globally asymptotically stable. Thus the effect of any disturbance will decay, and the pendulum will be restored to its vertical position. The feedback controller just designed and applied to (23). So the ultimate feedback controller, which is designed is given by the reference input,

   mg sin    ' (89)

or    mg sin   3m  2m .

From (89), it is clear that the system becomes an autonomous system after feedback linearization and stabilization. The torque  is an explicit function of the angle θ and its derivative, the angular velocity θ. The controller is a feedback controller. Because it takes measurements of θ and  and uses this information to adjust the motor torque in a way which is stabilizing, In Fig.27, for instance, if θ is non-zero, then the last term (proportional term) in (89) has the effect of forcing the motor to act in a direction opposite to the natural tendency to fall. The second term (derivative term) in (89) responds to the speed of the pendulum. It is worth noting that the feedback controller (89) has fundamentally altered the dynamics of the pendulum. With the controller in place, it is no longer an unstable nonlinear system. Indeed, in a different context, it was reported in the article [30], that it is possible to remove the effects of chaos with a suitable controller, even using a simple proportional control method. It is noted that this design procedure requires explicit knowledge of the model parameters (length, mass, etc), and if they are not known exactly, performance may be degraded. Similarly, unmodeled influences (such as friction, motor dynamics) also impose significant practical limitations. Ideally, one would like a design which is robust and tolerates these negative effects. The objective of this chapter is to make the deadbeat realization after the feedback linearization and stabilization. In this context the derivation of transfer function is essential to get the linear time invariant transient response of the system. B. Transfer Function after Linearization By feedback linearization, it is obvious that the dotted part has transfer function

T1 ( s) 

.. 1 as  '  m  ms2

(90)

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MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

and transfer function of the rest part is (91) ' T2 ( s)   3ms  2m  If it is assumed that that some disturbance force acting on the pendulum, then the linear transfer function with stability consideration 1 G (s) T (s)   1  G(s) H (s)

ms2 1

1 ms2

m(3s  2)



1 2 ms  3ms  2m

(92)

(93)

so the characteristic equation is ms2  3ms  2m  0

Fig. 27 represents the transfer function with stability consideration. The transient response of (93) is checked with the block diagram given in Fig. 28.The objective is to make the deadbeat realization of this transient response using SCT. In section III an additional signal has been used to compensate the output as a deadbeat form. The same technique will be used here for dead beat realization of the system dynamic of inverted pendulum. If m=1/30 gm =0.0333 gm is considered with the unit step input, then using (92) as a transfer function (after feedback linearization) with closed loop system the transient response of the whole system must be linear. The transient response of the system (92) is shown in Fig. 29. The linearized transfer function of the dynamic system of inverted pendulum is

.. I

mg sin   l mg sin   l

mg



Figure 25. Inverted pendulum or 1-link robot arm

T ( s) 

Figure 26. Torque calculation of the model of inverted pendulum

30 2 s  3s  2

(94)

The closed loop transfer function of the dynamics of inverted pendulum is

 ( s) 

T ( s) 30  1  T (s) H (s) s 2  3s  32

(95)

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MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

1 ms 2 

m3s  2

'

Figure 27. Feedback stabilization of pendulum

Figure 28. Transfer function with stability consideration

If y t   Y1 t   1.5t  0.5t 2 is considered as the polynomial curve for the deadbeat response y(t) in the interval 0  t  t 0 then the rise time t0 = 1.0 and y(t)=u(t)=1 for t>t0. The rise of the deadbeat system will improve for

considering

y(t )  Y2 (t )  1.7679t - 0.5t.2 in the interval 0  t  t 0 . Then the rise time t 0  0.707 and y(t)  u(t)  1 for t  t 0 . 2

Transient response

1.5 1 0.5 0 -0.5 -1 -1.5 -2

0

2

4

6

8

10

t in secs

Figure 29. Transient Response of Inverted Pendulum after Linearization 2

1.5

Dead beat response Y(t)

1.5

Corrected signal

1 0.5 0 -0.5 -1

1

0.5

0

-0.5

-1.5 -2

0

2

4

6

8

-1

10

t in secs

0

2

4

6

8

10

t in secs

Figure 30. The Corrected Signal for inverted pendulum for t0=1 sec

Figure 31. Deadbeat response of Inverted Pendulum for t0=1 sec

Table 1 represents the selected polynomial curve Y1(t) and Y2(t) and corresponding rise time for the system (92). In Fig. 30 and Fig. 31 the corrected signal and deadbeat response of (92) are shown respectively with

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MAYFEB Journal of Electrical and Computer Engineering Vol 2 (2017) - Pages 1-23

polynomial curve Y1(t). Similarly Fig. 32 and Fig. 33 represents the corrected signal and deadbeat response of (92) respectively with polynomial curve Y2(t). 2 1.5

Corrected signal

1 0.5 0 -0.5 -1 -1.5 -2

0

2

4

6

8

10

t in secs

Figure 32. The Corrected Signal for inverted pendulum for t0=0.707 VII.

Figure 33. Deadbeat response of Inverted Pendulum for t0=0.707

CONCLUSION AND FUTURE SCOPE

The work reported in this paper is mainly concerned with the realization of dead beat transient response of linear and nonlinear system using SCT scheme. It is pointed out that in some applications, such as biological control systems, it may not always be possible to incorporate a controller inside the system. Here the dead-beat control has been achieved by signal correction technique, which does not require any restriction on the system parameters and experimental data. This SCT scheme whatever applied in this work is very much suitable for the application in biological system. In this manuscript a corrected signal is generated through the state space equation and applied to the system through the feedback loop. This feedback is used to stabilize and regulate a system in the presence of disturbances and uncertainty. The main problem of control engineers is to design feedback controllers. In our presentation feedback linearization is applied to make the dynamic system of the inverted pendulum, a globally asymptotically stable. This SCT technique, which is considered here, can be applied to any higher order linear, nonlinear system for deadbeat realization without restriction of system parameters and experimental data after ensuring the stability of the system. However there is a huge scope also to explore the SCT based deadbeat control scheme to other nonlinear and time varying system, such as Industrial plants, Flight Control System, UPS Inverter, Rocket and Missile, Balancing Robots, Pitch control of an aircraft, etc..As deadbeat realization is applicable only for the linearized system, the nonlinear systems are approximated by linear models using suitable techniques known as feedback linearization and other linearization technique, which can introduce the global stability and robustness of the system. REFERENCES [1] Mandal A. K ―Introduction to Control Engineering- Modeling, Analysis and Design‖ New age International, New Delhi, Second Edition, 2012 [2] Brian D.O.Anderson, John B Moore, ―Optimal Control: Linear Quadratic Methods‖, Prentice HallInc, Englewood Cliffs, New Jersey, 2007 [3] Thomas Kailath,Ali, H. Sayed, BabakHassibi, ―Linear Estimation‖, Prentice Hall, ,2000 [4] Farhan A. Salem, ‖ Mechatronics Motion Control Design of Electric Machines for Desired Deadbeat Response Specifications Supported and Verified by New Matlab Built-In Function and Simulink Model‖, European Scientific Journal Volume.9, No.36, pp 357-386. 2013 [5] Acintya Das, Rajib Bag, And N. G. Nath, ―A Modification To Realize Dead-Beat Performance Of Control Systems—Signal Correction Technique‖, IEEE Transactions On Instrumentation And Measurement, Volume. 55, No. 5, pp: 1546–1550., 2006

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