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3.3 Phase trajectories of an autonomous second order system

Lecture



Consider a second-order autonomous system:



(3.74)   3.3 Phase trajectories of an autonomous second order system ,



with initial values   3.3 Phase trajectories of an autonomous second order system ,   3.3 Phase trajectories of an autonomous second order system . System characteristic equation



(3.75)   3.3 Phase trajectories of an autonomous second order system



has two real or complex-conjugate roots (poles of the system):



(3.76)   3.3 Phase trajectories of an autonomous second order system ,



whose location on the complex plane determines the type of transient processes



(3.77) y = y (y 0 ,   3.3 Phase trajectories of an autonomous second order system , t )



  3.3 Phase trajectories of an autonomous second order system



and dynamic properties of the system (see § 1.4.1).



Define state variables as phase variables:   3.3 Phase trajectories of an autonomous second order system ,   3.3 Phase trajectories of an autonomous second order system . The state-exit model takes the form:



(3.78)   3.3 Phase trajectories of an autonomous second order system ,



(3.79)   3.3 Phase trajectories of an autonomous second order system ,



(3.80)   3.3 Phase trajectories of an autonomous second order system ,



with initial values   3.3 Phase trajectories of an autonomous second order system ,   3.3 Phase trajectories of an autonomous second order system Eigenvalues ​​of the system matrix



  3.3 Phase trajectories of an autonomous second order system



coincide with the roots of the characteristic polynomial (3.75) p 1,2 .



Own vectors   3.3 Phase trajectories of an autonomous second order system 1.2 (see § 3.1.3) of the second-order system under consideration are found (provided that its poles are real) from the expression



  3.3 Phase trajectories of an autonomous second order system ,



those.



  3.3 Phase trajectories of an autonomous second order system



and the corresponding eigenspaces R 1,2 are represented by straight lines.



(3.81) x 2 = p 1,2 x 1 .



The equilibrium (steady) states ( x 1 *, x 2 * ) of the system (3.78), (3.79), (3.80) are found from the condition



(3.82)   3.3 Phase trajectories of an autonomous second order system ,   3.3 Phase trajectories of an autonomous second order system .



When a 2   3.3 Phase trajectories of an autonomous second order system 0 we find that the equilibrium position is the origin



(3.83) x 1 * = 0 , x 2 * = 0 ,



and for a 2 = 0 we find the set of equilibrium states (straight line)



(3.84) x 2 = 0.



Recall that the integral curve ( phase path) of the system under consideration is the hodograph of the state vector   3.3 Phase trajectories of an autonomous second order system =   3.3 Phase trajectories of an autonomous second order system when the parameter t changes , and the set of phase trajectories obtained for different initial conditions form its phase portrait (see § 3.1.2). Phase trajectories can be obtained experimentally or found analytically. In the latter case, use the following technique. Equations (3.78), (3.79) are written as



dx 1 = x 2 dt,



dx 2 = - (a 2 x 1 + a 1 x 2 ) dt.



After dividing the second expression by the first, we obtain the differential equation



(3.85)   3.3 Phase trajectories of an autonomous second order system .



The solution of this equation is sought in the form



(3.86) x 2 =   3.3 Phase trajectories of an autonomous second order system one   ( x 1 )



and determines the integral (phase) trajectory of the considered



systems on the plane r 2 .



Consider transients corresponding to different values ​​of the roots of the characteristic equation (poles of the system (3.78) - (3.80)).



1. For unequal real roots (Fig. 3.4, 3.5)



(3.87) p 1,2 =   3.3 Phase trajectories of an autonomous second order system 1.2 =   3.3 Phase trajectories of an autonomous second order system



equation (3.74) has a solution



(3.88)   3.3 Phase trajectories of an autonomous second order system ,



which corresponds to the aperiodic process (see § 2.2.2).



Provided that a 1 > 0 and a 2 > 0,



Re p 1,2 =   3.3 Phase trajectories of an autonomous second order system 1.2 <0,



(Fig. 3.4). In this case, there is a damped transient, the condition



  3.3 Phase trajectories of an autonomous second order system



Fig. 3.4

Fig. 3.5



(3.89)   3.3 Phase trajectories of an autonomous second order system



and phase trajectories of the system with t   3.3 Phase trajectories of an autonomous second order system   3.3 Phase trajectories of an autonomous second order system converge to the equilibrium position O (Fig. 3.6), which is called a stable node . (A system of this kind belongs to the class of asymptotic-stable systems [11,12]). The system has two proper (invariant) subspaces R 1 and R 2 , on which the solutions of equation (3.74) are written as



  3.3 Phase trajectories of an autonomous second order system



Fig. 3.6

Fig. 3.7



(3.90)   3.3 Phase trajectories of an autonomous second order system



and



(3.91)                         3.3 Phase trajectories of an autonomous second order system ,



those. the dynamics on the eigenspaces corresponds to the behavior of the first order system.



Provided that a 1 > 0 and a 2 = 0, we get



Re p 1 =   3.3 Phase trajectories of an autonomous second order system <0, p 2 = 0,



(Fig. 3.5). Phase trajectories of the system (Fig. & 3. 7) with t   3.3 Phase trajectories of an autonomous second order system   3.3 Phase trajectories of an autonomous second order system converge to the set of equilibrium states (straight line R 0 ) described by equation (3.84). This set is its own subspace of the system. (A system of this kind belongs to the class of stable, or neutral-stable, systems, see [10,12])



Provided that a 1 <0 and a 2 > 0,



Re p 1 =   3.3 Phase trajectories of an autonomous second order system 1 > 0, Re p 2 =   3.3 Phase trajectories of an autonomous second order system 2 <0,



  3.3 Phase trajectories of an autonomous second order system



Fig. 3.8 Figure 3.9



(Fig. 3.8). In this case, there is a divergent transition process. Phase trajectories of the system with t   3.3 Phase trajectories of an autonomous second order system   3.3 Phase trajectories of an autonomous second order system diverge (Fig. 3.10):



(3.92)   3.3 Phase trajectories of an autonomous second order system



with the exception of trajectories starting on the straight line R 2 , for which the limit relation (3.89) is satisfied. (A system of this kind belongs to the class of unstable systems, see [10,12]) The equilibrium position of the system (point O ) is called a saddle point (saddle) . The system has two proper (invariant) subspaces R 1 and R 2 , on which the solutions (3.74) are written in the form (3.90) or (3.91).



  3.3 Phase trajectories of an autonomous second order system



Fig. 3.10 Figure 3.11







Provided that a 1 <0 and a 2 <0,



Re p 1 =   3.3 Phase trajectories of an autonomous second order system 1 > 0, Re p 2 =   3.3 Phase trajectories of an autonomous second order system 2 > 0,



(Fig. 3.9). In this case, there is a diverging transition process and all phase trajectories (Fig. 3.11) of the system at t   3.3 Phase trajectories of an autonomous second order system   3.3 Phase trajectories of an autonomous second order system diverge (performed (3.92)). The equilibrium position of the system (point O ) is called an unstable node (and the system is also unstable). The system also has two proper (invariant) subspaces R 1 and R 2 .



2. If   3.3 Phase trajectories of an autonomous second order system , then the system has equal real poles (fig. 3.12 - 3.14)



p 1,2 =   3.3 Phase trajectories of an autonomous second order system =   3.3 Phase trajectories of an autonomous second order system



and the solution of equation (3.74) takes the form:





  3.3 Phase trajectories of an autonomous second order system



Fig. 3.12 Figure 3.13 Figure 3.14



y = (C 1 + C 2 t)   3.3 Phase trajectories of an autonomous second order system ,



corresponding to the aperiodic process (see § 2.2.2).



Provided that a 2 > 0 (and a 1 > 0),



Re p 1,2 =   3.3 Phase trajectories of an autonomous second order system <0,



  3.3 Phase trajectories of an autonomous second order system



Fig. 3.15 Figure 3.16 Figure 3.17



(fig. 3.12). In this case, a decaying transient takes place, the limiting relation (3.89) is fulfilled, phase trajectories at t   3.3 Phase trajectories of an autonomous second order system   3.3 Phase trajectories of an autonomous second order system converge to the equilibrium position (stable node) O (Fig. 3.15) and the system is asymptotically stable. The eigenspaces of the systems R 1 and R 2 coincide.



Provided that a 1 = a 2 = 0, we get



p 1 = p 2 = 0,



(Fig. 3.13) and the divergent transition process. Phase trajectories of the system (Fig. 3.16) with t   3.3 Phase trajectories of an autonomous second order system   3.3 Phase trajectories of an autonomous second order system go to infinity, with the exception of trajectories starting at the set of equilibrium states (line R 0 ), described by the equation x 2 = const, and the system is unstable.



Provided that a 2 <0 and a 1 <0,



Re p 1,2 =   3.3 Phase trajectories of an autonomous second order system > 0,



rice 3.12) ,. proper subspaces of the system coincide. In this case, the limiting relation (3.89) holds, phase trajectories at t   3.3 Phase trajectories of an autonomous second order system   3.3 Phase trajectories of an autonomous second order system diverge. The equilibrium position O is an unstable O node (Fig. 3.17) and the system is unstable.



3 If performed   3.3 Phase trajectories of an autonomous second order system , then the system has complex conjugate poles (Fig. 3.18-3.20)



  3.3 Phase trajectories of an autonomous second order system   3.3 Phase trajectories of an autonomous second order system



  3.3 Phase trajectories of an autonomous second order system



Fig. 3.18 Figure 3.19 Figure 3.20



and the solutions of equation (3.74) take the form



  3.3 Phase trajectories of an autonomous second order system



which corresponds to the oscillatory process (see § 2.2.2).



The system with complex poles considered here does not have its own subspaces.



Provided that a 1 > 0 and a 2 > 0,



Re p 1,2 =   3.3 Phase trajectories of an autonomous second order system <0,



(fig. 3.18). In this case, there is a damped oscillatory transient process. Running (3.89), phase trajectories of the system at t   3.3 Phase trajectories of an autonomous second order system   3.3 Phase trajectories of an autonomous second order system converge to the equilibrium position O (Fig. 3.21), which is called a stable focus, and the system is asymptotically stable.



Provided that a 2 = 0, Re p 1,2 = 0, the system has pure imaginary roots



p 1,2 = - j   3.3 Phase trajectories of an autonomous second order system



(fig. 3.22) is also called a linear oscillator (see § 2.3). In this case, a continuous oscillation process takes place. The phase trajectories of the system are represented by closed concentric curves (elliptical orbits), and the system is (neutral) stable. The equilibrium position of the system (point O ) is called the center



  3.3 Phase trajectories of an autonomous second order system



Fig. 3.21 Pic. 3.22 Figure 3.23



Provided that a 1 <0 and a 2 <0,



Re p 1,2 =   3.3 Phase trajectories of an autonomous second order system > 0,



(fig. 3.20). In this case, there is a diverging oscillatory transition process. Phase trajectories of the system with t   3.3 Phase trajectories of an autonomous second order system   3.3 Phase trajectories of an autonomous second order system diverge from the equilibrium position O (Fig. 3.23), which is called an unstable focus , and the system is unstable.





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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control