DSAS_8_Phase Plane Method下载_在线阅读_13

is_911566

暂无简介

DSAS_8_Phase Plane Method NCTU Department of Electrical and Computer Engineering Senior Course By Prof. Yon-Ping Chen 8-1 8. Phase Plane Method A. Phase Plane B. Natural Modes of LTI Systems C. Phase Portraits Phase plane method is an important graphical methods to deal with...

NCTU Department of Electrical and Computer Engineering Senior Course By Prof. Yon-Ping Chen 8-1 8. Phase Plane Method A. Phase Plane B. Natural Modes of LTI Systems C. Phase Portraits Phase plane method is an important graphical methods to deal with problems related to a second-order autonomous system. A. Phase Plane First, let’s introduce the basic concepts of the phase plane by a second-order autonomous system which is mathematically expressed as below: ( ) ( ) ( )( )txtxftx 2111 ,=& , ( ) 101 0 xx = (8-1) ( ) ( ) ( )( )txtxftx 2122 ,=& , ( ) 202 0 xx = (8-2) without any inputs. It is known that the system states x1(t) and x2(t) are uniquely determined by initial values ( ) 101 0 xx = and ( ) 202 0 xx = . Clearly, different initial conditions ( ) 101 0 xx = and ( ) 202 0 xx = will result in different states x1(t) and x2(t) for t>0. If we adopt the values of the two state variables as two axes, then a coordinate is formed in Figure 8-1, called the phase plane. We can draw all the points (x1(t), x2(t)) for t≥0 to obtain a trajectory with respect to specified initial conditions ( )01x and ( )02x . Figure 8-1 shows an example of the trajectory (x1(t),x2(t)) for 0 By Prof. Yon-Ping Chen 8-2 Since each trajectory is uniquely determined by a set of initial conditions, that means no intersection exists among all the trajectories, except at a singular point (x1e,x2e) satisfying ( ) 01 =tx& and ( ) 02 =tx& for all t, i.e., ( )txxf ee ,, 2110 = (8-3) ( )txxf ee ,, 2120 = (8-4) In fact, (x1e,x2e) is an equilibrium point and not necessary to be unique. A set of trajectories obtained from different initial conditions is called a phase portrait, as shown in Figure 8-2, where P is an equilibrium point or a singular point. B. Natural Modes of LTI Systems The natural modes reflects the intrinsic properties of systems which are not excited by any external inputs. In other words, the natural modes are the system behavior caused only by its initial conditions. Let’s employ the simplest 2nd order dynamic system as an example to explain how to determine the natural modes of an LTI system. Consider the following 2nd order LTI system without any input: ( ) ( ) ( )txatxatx 2121111 +=& , x1(0)=x10 (8-5) ( ) ( ) ( )txatxatx 2221212 +=& , x2(0)=x20 (8-6) or in a matrix form          ⋅= tt xAx& , 0 0xx = (8-7) x1 x2 Figure 8-2 t=0 P NCTU Department of Electrical and Computer Engineering Senior Course By Prof. Yon-Ping Chen 8-3 where       = 2221 1211 aa aa A , ( ) ( )     =     0 0 0 2 1 x x x and       = 20 10 0 x x x . Clearly, the origin (0,0) in the phase plane is an equilibrium point since the truth of 0x =    t leads to 0x =    t& . From (1-8), the state vector can be solved as 0xx A tet =     (8-8) with te A defined as LL !n t ! t te nn t AAAIA ++++= 2 22 (8-9) a form similar to LL !! n xx xe n x ++++= 2 1 2 . It is known that any square matrix A can be transformed into a matrix J in diagonal form or Jordan form, expressed as 1− = VJVA (8-10) where V is invertible. It is easy to show that 1− = VVJA nn n=0,1,2,… (8-11) There are three typical forms of J, denoted as       = 2 1 0 0 λ λJ ,       = r r λ λ 0 1J or       = ∗ c c λ λ 0 1J (8-12) where the disgonal terms are the eigenvalues of A and can be solved from its characteristic equation 0 2221 1211 = −− −− =− aa aa λ λλ AI (8-13) That means they are the solutions of ( ) 02112221122112 =−++− aaaaaa λλ (8-14) and both of them may be real or complex in conjugate and may be distinct or repeated. Accordingly, there are three cases of the eigenvalues: Case-1 distinct real eigenvalues, 21 λλ ≠ Case-2 repeated real eigenvalues, rλλλ == 21 Case-3 complex eigenvalues in conjugate, ωαλ jc += and ωαλ jc −=∗ . It is known that for an eigenvalue λi there exists at least one eigenvector vi such that iii vAv λ= (8-15) NCTU Department of Electrical and Computer Engineering Senior Course By Prof. Yon-Ping Chen 8-4 For Case-1, based on (8-15) we have    = = 222 111 vAv vAv λ λ (8-16) which results in [ ] [ ] 43421 4342143421 JV vv V vvA       = 2 1 2121 0 0 λ λ (8-17) Note that J is the first typical form in (8-12) and V is invertible and called the eigenvectormatrix since it contains all the eigenvectors of A. Clearly, (8-17) can be rearranged as 1 −= VJVA , a form given in (8-10). Besides, we have 1 2 11 0 0 −−       == VVVVJA n n nn λ λ (8-18) Hence, the state vector in (8-8) is obtained as 0 022 2 2 22 1 1 0 22 0 22 0 2 1 0 0 2 10 0 2 1 2 2 xVV xVV xVJJJIV x AAAI xx 1 1 1 A − − −            =             +++ +++ =       ++++=       ++++= = t t nn nn t e e ! t t ! t t !n t ! t t !n t ! t t et λ λ λλ λλ L L LL LL (8-19) or in the following form ( ) tt eetx 21 12111 λλ αα += (8-20) ( ) tt eetx 21 22212 λλ αα += (8-21) where α11, α12, α21 and α22 depend on x1(0) and x2(0). The functions te 1λ and te 2λ are called the natural modes of this system. For Case-2 of repeated eigenvalue λr, we may obtain one or two eigenvectors correspondingly. Let’s assume only one eigenvector is found, then we have rrr vAv λ= (8-22) NCTU Department of Electrical and Computer Engineering Senior Course By Prof. Yon-Ping Chen 8-5 It is known that we can create a new vector rv′ which satisfies rrrr vvvA +′=′ λ (8-23) and most importantly rv and rv′ are independent. As a result, wwe attain [ ] [ ] 43421 4342143421 JV vv V vvA       ′=′ r r rrrr λ λ 0 1 (8-24) Note that J is the second typical form in (8-12) and V is invertible since rv and rv′ are independent. Clearly, (8-24) can be rearranged as 1 −= VJVA , same as (8-10). Besides, we have 1 1 1 0 − − −       == VVVVJA n r n r n rnn n λ λλ (8-25) Hence, the state vector in (8-8) is obtained as 0 022 32 2 22 0 22 0 22 0 0 2 10 2 0 2 1 2 2 xVV xVV xVJJJIV x AAAI xx 1 1 1 A − − −            =             +++ +++++++ =       ++++=       ++++= = t tt r r r r r r nn nn t r rr e tee ! t t ! t tt ! t t !n t ! t t !n t ! t t et λ λλ λλ λλλλ L LL LL LL (8-26) or in the following form ( ) tt rr teetx λλ βγ 111 += , (8-27) ( ) tt rr teetx λλ βγ 222 += , (8-28) where γ1, γ2, β1 and β2 depend on x1(0) and x2(0). The functions treλ and trteλ are called the natural modes of this system. For Case-3 of complex eigenvalues in conjugate, we will skip the derivation and just show the resulted state variables which are ( ) tsineBtcoseAtx tt ωω αα 111 += (8-29) ( ) tsineBtcoseAtx tt ωω αα 222 += (8-30) NCTU Department of Electrical and Computer Engineering Senior Course By Prof. Yon-Ping Chen 8-6 where A1, A2, B1 and B2 depend on x1(0) and x2(0). The functions tcose t ωα and tsine t ωα are called the natural modes of this system. Note that if α=0, i.e., the eigenvalues are pure imaginary, then the natural modes are tcosω and tsinω . It is clear that all the natural modes depend on the eigenvalues of the system matrix A. If an eigenvalue is located on the left half complex plane, then the natural mode corresponding this eigenvalue will converge to 0 as t→∞. On the other hand, if an eigenvalue is located on the right half complex plane, then the related natural mode will increase to ∞ as t→∞. As for the eigenvalue on the imaginary axis, its natural mode will oscillate. C. Phase Portraits Now, let’s use some examples to draw phase portraits, a set of trajectories, and discuss their related properties. Actually, different natural modes will result in different trajectories and different phase portraits as well. Consider the second-order LTI system in canonical form, which is described as below: ( ) ( )txtx 21 =& , x1(0)=x10 (8-31) ( ) ( ) ( )txatxatx 2221212 +=& , x2(0)=x20 (8-32) The system matrix is       = 2221 10 aa A and the characteristic equation is 0 1 2122 2 2221 =−−= −− − =− aa aa λλλ λλ AI (8-33) whose roots are 2 4 21 2 2222 21 aaa +± =λλ , (8-34) If 04 21 2 22 >+ aa , then λ1 and λ2 are real and distinct. Without loss of generality, assume λ1>λ2, then there are three cases listed as below: I. λ2<λ1<0, i.e., 04 21222 >+ aa , a22<0 and a21<0. II. λ1>λ2>0, i.e., 04 21222 >+ aa , a22>0 and a21<0. NCTU Department of Electrical and Computer Engineering Senior Course By Prof. Yon-Ping Chen 8-7 III. λ1>0>λ2, i.e., 04 21222 >+ aa and a21>0. Their phase portraits are depicted in Figure 8-3(a)(b)(c), respectively. From (8-20), the first state variable ( )tx1 is ( ) tt eetx 21 12111 λλ αα += (8-35) and from (8-31), the second state variable is ( ) ( ) tt eetxtx 21 21211112 λλ λαλα +== & (8-36) For some initial conditions x1(0) and x2(0), the coefficient α12 in (8-35) may be equal to 0, then we have ( ) tetx 1111 λα= (8-37) ( ) ( )txetx t 111112 1 λλα λ == (8-38) which implies ( ) ( ) 0112 =− txtx λ (8-39) i.e., the related trajectory is the straight line L1 shown in Figure 8-3(a)(b)(c). Similarly, if initial conditions result in α11=0, then ( ) tetx 2121 λα= (8-40) ( ) ( )txetx t 122122 2 λλα λ == (8-41) which implies ( ) ( ) 0122 =− txtx λ (8-42) i.e., the related trajectory is the straight line L2 shown in Figure 8-3(a)(b)(c). In Case-I with λ2<λ1<0, its phase portrait is depicted in Figure 8-3(a). As t→∞, due to the fact that 021 →>> tt ee λλ and 02 →teλ , all the trajectories except L2 in the phase portrait must satisfy ( ) tetx 1111 λα→ (8-43) ( ) ( )txetx t 111112 1 λλα λ =→ (8-44) i.e., ( ) ( ) 0112 →− txtx λ . In other words, all the trajectories except L2 in the phase portrait will approach L1 as t→∞. Clearly, it is a stable system since the system state will converge to the equilibrium point (0,0) as t→∞. NCTU Department of Electrical and Computer Engineering Senior Course By Prof. Yon-Ping Chen 8-8 In Case-II and Case III, their phase portraits are depicted in Figure 8-3(b) and Figure 8-3(c). As t→∞, due to the fact that tt ee 21 λλ >> and ∞→te 1λ , all the trajectories except L2 in the phase portrait must satisfy ( ) tetx 1111 λα→ (8-45) ( ) ( )txetx t 111112 1 λλα λ =→ (8-46) i.e., ( ) ( ) 0112 →− txtx λ . In other words, all the trajectories except L2 in the phase portrait will approach L1 as t→∞. Clearly, it is an unstable system since the system state will move to (∞, ∞) or (−∞, −∞) as t→∞, except L2 in Case-II. However, Case-II is still treated as unstable because any extremely small disturbance will drive the system to leave L2 and move to (∞, ∞) or (−∞, −∞). If 04 21 2 22 =+ aa , then from (8-34) we have 2 22 21 a == λλ . There are two cases listed as below: IV. λ2=λ1<0, i.e., 04 21222 =+ aa and a22<0. x1 (a) (b) Figure 8-4 x1 x2 x2 x1 x2 x1 (a) (b) (c) Figure 8-3 L2 L1 x1 x2 L1 L2 x2 L1 L2 NCTU Department of Electrical and Computer Engineering Senior Course By Prof. Yon-Ping Chen 8-9 V. λ1=λ2>0, i.e., 04 21222 =+ aa and a22>0. Both phase portraits are depicted in Figure 8-4(a)(b), respectively. In fact, they are special cases of Case-I and Case-II when L1 and L2 are merged together. From the phase portraits, it is easy to conclude that Case-IV is stable and Case-V is unstable. If 04 21 2 22 <+ aa , then λ1 and λ2 are complex in conjugate. There are two cases listed as below: VI. λ2,λ1=±jω where ω>1, ω=1 or ω<1. VII. λ2,λ1=α±jω where α>0 or α<0. Their phase portraits are depicted in Figure-5(a)(b)(c) and Figure-6(a)(b). For Case-VI, from (8-29) with α=0, the first state variable is ( ) ( )θωωω +=+= tcosCtsinBtcosAtx 111 (8-47) and from (8-31) the second state variable is ( ) ( ) ( )θωω +−== tsinCtxtx 12 & (8-48) where C is determined by the initial conditions x1(0) and x2(0). Clearly, it is an oscillatory system. From (8-47) and (8-48), we have ( ) ( ) 22 2 22 1 C tx tx =+ ω (8-49) whose phase portraits are ellipses or circles, shown in Figure 8-5(a)(b)(c) respectively for ω>1, ω=1 and ω<1. Note that the arrow in these figures is pointing from left to right in the region x2(t)>0 since ( ) ( ) 021 >= txtx& , i.e., ( )tx1 is increased as t increases, and from right to left in the region x2(t)<0 since ( ) ( ) 021 <= txtx& , i.e., ( )tx1 is decreased as t increases. While passing through the ( )tx1 axis, i.e., ( ) ( ) 021 == txtx& , the arrow goes down or up vertically. x1 x2 x1 x2 x1 x2 (a) (b) (c) Figure 8-5 NCTU Department of Electrical and Computer Engineering Senior Course By Prof. Yon-Ping Chen 8-10 The above description of the arrow direction is also suitable for Case-I to Case-V. For Case-VII, from (23), the first state variable is ( ) ( ) ( )θωωω αα +=+= tcoseCtsinBtcosAetx tt 111 (8-50) and from (8-31) the second state variable is ( ) ( ) ( ) ( ) ( )θω θωαθωω α αα ′+′= +++−== tcoseC tcoseCtsinCetxtx t tt 12 & (8-51) Clearly, it is a divergent oscillatory system for α>0 as shown in Figure 8-6(a) and an attenuated oscillatory system for α<0 as shown in Figure 8-6(b). Unlike an LTI system, the phase portrait of a nonlinear system is often more complicated since it may have multiple equilibrium points or limit cycles. However, the local behavior of a continuous nonlinear system around an equilibrium point can be determined by linearization which results in an approximted LTI system near to the equilibrium point. If the approximated LTI system is stable then the equilibrium point is stable, otherwise it is unstable. Now, consider a nonlinear system and determine its stability by the use of phase plane method. Let the state equations be ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( )  −++−= −++= 1 1 2 2 2 1212 2 2 2 1121 txtxtxtxtx txtxtxtxtx & & (8-52) If ( ) ( ) 012221 =−+ txtx , then it is easy to solve the state variables as ( ) ( )θ+= tcostx1 and ( ) ( )θ+−= tsintx2 . In other words, if the system is initially located on the circle ( ) ( ) 012221 =−+ txtx , then the system state will keep in a circular trajectory. While the initial condition is a little deviated from the circle, for example ( ) ( ) ε=−+ 12221 txtx , x1 (a) (b) Figure 8-6 x1 x2 x2 NCTU Department of Electrical and Computer Engineering Senior Course By Prof. Yon-Ping Chen 8-11 its behavior is then governed by ( ) ( ) ( ) ( ) ( ) ( )  +−= += txtxtx txtxtx 212 211 ε ε & & (8-53) whose system matrix is       − ε ε 1 1 . Correspondingly, its characteristic equation is 0122 =+− ελλ (8-54) and the eigenvalues are 1 2 442 2 21 j±≈ −± = ε εελλ , . Clearly, for ε>0 the system is in the region outside the circle and becomes unstable, i.e., it moves away the circle to the infinity. For ε<0 the system is in the region inside the circle and becomes stable, i.e., it moves away the circle to the origin. Figure 8-7 shows the phase portrait including a limit cycle ( ) ( ) 012221 =−+ txtx . Next, let’s consider a variable structure system, described by the following state equations ( ) ( ) ( ) ( ) ( )( )( ) ( )  +−= = txtxtxsigntx txtx 1212 21 5152 ..& & (8-55) It is obvious that there are two structures of the system, one for ( ) ( ) 021 >txtx and the other for ( ) ( ) 021 txtx , i.e., in the first and third quadrants of the phase plane, the system is governed by x1 x2 ( ) ( ) 012221 =−+ txtx Figure 8-7 NCTU Department of Electrical and Computer Engineering Senior Course By Prof. Yon-Ping Chen 8-12 ( ) ( ) ( ) ( )  −= = txtx txtx 12 21 4 & & (8-56) which performs as an ellipse. While ( ) ( ) 021 > % key in the following instructions >> [t,x]=ode45(@vss,[0:0.01:10],[10 0]) >> plot(x(:,1),x(:,2)); xlabel(‘x1’); ylabel(‘x2’); grid -5 0 5 10 -10 -5 0 5 x1 x 2 ========================================== It can be concluded that the variable structure system is stable, even though both of its sub-systems are unstable. Problems P.8-1 Consider the following system: ( ) ( )txtx 21 =& , x1(0)=5 ( ) ( ) ( )txtxtx 212 4 −−=& , x2(0)=0 NCTU Department of Electrical and Computer Engineering Senior Course By Prof. Yon-Ping Chen 8-13 Draw the trajectory on the phase plane from t=0 to 10 in MATLAB.

本文档为【DSAS_8_Phase Plane Method】，请使用软件OFFICE或WPS软件打开。作品中的文字与图均可以修改和编辑，图片更改请在作品中右键图片并更换，文字修改请直接点击文字进行修改，也可以新增和删除文档中的内容。

DSAS_8_Phase Plane Method

热门搜索

历史搜索