Б. О. Джолдошева из Института автоматики и информационных технологий нан кр, г. Бишкек; «Cинтез кибернетических автоматических систем с использованием эталонной модели»
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2. Second order systemsA. Integrators in series. Let us consider a control plant presented by two integrators connected in series, as shown in Fig. 1: Figure 1. “Integrators in series” structure. where T1 and T2 are the parameters of integration. The structure of several integrators (more than 2 integrators) is famous of its instability, i.e. no one linear controller can provide the stability to such system and more over with uncertainly changeable parameters [8, 9]. The example of two integrators in series allows us to see the advantages of using non-linear catastrophe as controller. Let us choose a feedback control law as following form: , (1) and in order to study stability of the system let us suppose that there is no input signal in the system (equal to zero). Hence, the system with proposed controller can be presented as: (2) . The system (2) has following equilibrium points , ; (3) , . (4) Stability conditions for equilibrium point (3) obtained via linearization are (5) Stability conditions of the equilibrium point (4) are (6) By comparing the stability conditions given by (5) and (6) we find that the signs of the expressions in the second inequalities are opposite. Also we can see that the signs of expressions in the first inequalities can be opposite due to squares of the parameters k1 and k3 if we properly set their values. Let us suppose that parameter T1 can be perturbed but remains positive. If we set k2 and k3 both negative and then the value of parameter T2 is irrelevant. It can assume any values both positive and negative (except zero), and the system given by (2) remains stable. If T2 is positive then the system converges to the equilibrium point (3) (becomes stable). Likewise, if T2 is negative then the system converges to the equilibrium point (4) which appears (becomes stable). At this moment the equilibrium point (3) becomes unstable (disappears). Let us suppose that T2 is positive, or can be perturbed staying positive. So if we can set the k2 and k3 both negative and then it does not matter what value (negative or positive) the parameter T1 would be (except zero), in any case the system (2) will be stable. If T1 is positive then equilibrium point (3) appears (becomes stable) and equilibrium point (4) becomes unstable (disappears) and vice versa, if T1 is negative then equilibrium point (4) appears (become stable) and equilibrium point (3) becomes unstable (disappears). Results of MatLab simulation for the first and second cases are presented in Fig. 2 and 3 respectively. In both cases we see how phase trajectories converge to equilibrium points and. жүктеу/скачать 28,25 Mb. Достарыңызбен бөлісу:
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