Abstract - A variable structure oscillator, which ... where ω is circular frequency of sinusoidal self-oscil- ... this paper, the above mentioned idea will be used in.
SYNTHESIS OF HARMONIC OSCILLATOR WITH SLIDING MODE ^. Milosavljevi}, University of Ni{, Faculty of Electronic Engineering P. Radu-Emil, Technical University of Temisoara, Faculty of Automation and computers Abstract - A variable structure oscillator, which generates an ideal harmonic signal when it is in the sliding mode on the limit cycle in the state space, has been proposed. The sinusoidal responses are robust to parameter perturbation and external disturbances. 1. INTRODUCTION The sinusoidal signal generators are widely implemented as devices in development and research laboratories, production of radio and TV sets etc. The different approaches are used in the design of these generators [1]: the generators with Win's bridge, with amplitude limiter and automatic gain regulation, with phase shift etc. The simple scheme of the sinusoidal signal generator , which can be realized by modern operational amplifier and which is well known to engineers who are familiar with analog technique, is the scheme which is modeled by the following equations.
x&1 = x 2
x& 2 = −ω 2 x1
(1)
where ω is circular frequency of sinusoidal self-oscillations which present the solution of the system (1). The analog model of sinusoidal oscillator is given in Fig 1.
x&1 = x 2 x& 2 = 2ζω (1 − µx12 ) x 2 − ω 2 x1
(2)
The equation (2) is the mathematical model of the oscillator with the electronic tube. The design goal (the choice of parameters ζ and µ for the given ω) is to bring the coordinate states x1 and x2 from any initial state to the limit cycle. Harmonic oscillations will exist if the limit cycle is really circle or ellipse. However, the form of that limit cycle is sensitive to parameters ζ and µ, so the harmonic distortions appear. In 1987., Sira-Ramirez [2] forced Van der Pol oscillator to be ideal harmonic oscillator by introducing the additional control based on the theory of variable structure control with the sliding mode. In this paper, the above mentioned idea will be used in the synthesis of quality harmonic oscillator based on the model given in Fig. 1. The proposed solution is simpler than [2] in the realization by using the analog elements - operational amplifiers. In order to clearly understand the contribution of the proposed solution, it is necessary to briefly present the basic idea of harmonic oscillator proposed in [2]. 2. SIRA-RAMIREZ OSCILLATOR If the switching control signal u is introduced into the Van der Pol oscillator (2), the following mathematical model is obtained:
x&1 = x 2
Fig. 1 The analog model of sinusoidal oscillator
Such a generator does not practically give good results without the implementation of the additional elements for stabilization of amplitude and frequency. Therefore, the other solutions have been developed and they are mentioned above. In the control system theory, Van der Pol equation is very often used as the example of the system with the limit cycle in the state space. This equation in the state space has the following form [2, 3]:
x& 2 = 2ζω (1 − µx12 ) x 2 u − ω 2 x1 , u + = +1 for g > 0, u= − u = -1 for g < 0, g = ω 2 x12 + x 22 − ω 2 r 2 ,
(3)
(4) (5)
where relation g = 0 describes the limit cycle in the state space (x1, x2) on which the sliding mode should occurs. Ramirez has showed that the existence conditions of the sliding mode are as follows:
2ζω (1 − µx12 ) x 22 u + < 0,
(6)
2ζω (1 − µx12 ) x 22 u − > 0.
The analog model of Sira-Ramirez harmonic oscillator is shown in Fig. 2.
3. A SYNTHESIS OF A NEW TYPE OSCILLATOR
It is known from the variable structure control theory that the sliding mode occurs on the given switching lines where the phase trajectories face to each other. Sira-Ramirez has used this for establishing the sliding mode on the circle g = 0. We will do the same but we will start from equation of oscillator (1). Let us introduce the control so the system is described by:
x&1 = x 2 , x& 2 = −bx1 − x 2 u , b > 0,
Fig. 2 The analog model of Sira-Ramirez harmonic oscillator
In Fig. 3, the phase portrait and the time responses of Sira-Ramirez oscillator for µ=ζ=ω=1 are given.
1,5 1,0
(8)
and it can be easily obtained by implementation of the Lyapunov function. Time derivative of g along the trajectories of system structure motion is:
0,0
x2
where the control u is given by the relation (4). The goal is to choose the parameter b so the sliding mode occurs on the curve g = 0, which presents the phase trajectory of the harmonic oscillator. It is clear that we have here two linear structures for which the phase portraits have the unstable focus for u = u+ = +1 and the stable focus or node for u = u- = -1. Accordingly, there are possibility for establishing the sliding mode on the curve g = 0, if the adequate conditions would be ensured, which will be shown on the basis of variable structure control theory. The basic condition, which should be ensured for the existence of the sliding mode, is well known in the literature:
gg& < 0 ,
0,5
(7)
-0,5 -1,0
g& = 2ω 2 x1 x&1 + 2 x2 x&2 = 2ω 2 x1 x2 + 2 x2 (−ux2 − bx1 )
-1,5 -2,0
-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
2,0
= 2ω 2 x1 x2 − 2ux22 − 2bx1 x2
x1 2,0
x1(t)
1,5
1,0
0,5
0,0
-0,5
-1,0 0
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30
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50
time, s
Fig. 4 The analog realization of the proposed oscillator
Fig. 3 The phase portrait and the time responses of Sira-Ramirez oscillator.
It can be seen from these relations that the sliding mode can be ensured if b = ω2 because g>0 and u =+1 implies g& < 0 while g < 0 and u = -1 implies g& > 0 .
The analog model of the proposed solution of the harmonic oscillator is given in Fig. 4. Based on the analog model, someone can conclude that the proposed harmonic oscillator is simpler, because it has less elements and that it has the same quality as the oscillator designed on the basis of Van der Pol equation with the implementation of the sliding mode. The equality of quality originates from the fact that the equations of the sliding curve are formed in the same way, and because the sliding mode is ensured in the both cases. x2
1,5
1,0
0,5
x1
0,0
-0,5
-1,0
-1,5
It can be seen from the phase portraits and the time responses that the properties of discussed oscillators differ a little in the form of the phase trajectories before reaching the limit circle (the reaching mode), and that the time responses completely fits to each others after the transient mode. In such a way, the harmonic oscillator is obtained by the simpler realization. 4. CONCLUSION It has been shown in the paper that the quality harmonic oscillator can be obtained by combining the linear structures and the nonlinear control based on the variable structure control theory. In practical realization, the high frequency oscillations so-called chattering can occur which is caused by the nonidealities due to the switching elements, when the ideal sliding mode is no more possible. Next our resarch will be directed to the solwing of this problem by using so-colled reaching law approach [5]. REFERENCES
-2,0
-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
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x1(t) 1,5
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[1] T. Brodi}, Analog Integrated Electronics, "Svjetlost", Sarajevo [2] H. Sira-Ramirez, "Harmonic response of variable structure - controlled Van der Pol oscillators", IEEE Trans. on Circuit and Systems, CAS-34, No. 1, January 1987., pp. 103 - 106 [3] J. C. Hsu, A. U. Meyer, Modern Control Principles and Applications, 1972., (in Russian) [4] V. I. Utkin, "Variable structure control systems with sliding modes", IEEE Trans. On Automatic Control, Vol. AC-22, pp. 212-222, April 1997. [5] J. Y. Hang et all.:"Variable Structure Control: A Survey", IEEE Trans. On Ind. Electronics, Vol. 40, No 1, pp.2-22, Febr. 1993.
time, s
Fig. 5 The phase portrait and the time responses of the proposed oscillator.
It is necessary to point out the fact that, when x2 = 0, the reaching conditions of the sliding mode are not fulfilled. Then the control is u = 0. The system will be forced to move by the acting of the term ω2x1.The system state will immediately leave the sector x2 = 0. Accordingly, in the given case, someone can claim that the system will reach the given limit cycle (the sliding curve) from any initial state, excluding the point x1 = x2 = 0 only. In Fig. 5, the phase portrait and the time responses of the proposed harmonic oscillator are given.
SINTEZA HARMONIJSKOG OSCILATORA SA KLIZNIM RADNIM RE@IMOM ^edomir Milosavljevi} Precup Radu-Emil
Sadr`aj - Prikazan je oscilator promenljive strukture za generisanje idealnih harmonijskih signala, kada se kretanje odvija u kliznom re`imu na grani~nom krugu, u faznom prostoru stanja. Sinusoidalni odzivi su robusni na varijacije parametara i spolja{nje poreme}aje.
