Local Stabilization of Nonlinear Control Systems: Linear Approximation & Zero Dynamics, Exercises of Nonlinear Control Systems

Download Local Stabilization of Nonlinear Control Systems: Linear Approximation & Zero Dynamics and more Exercises Nonlinear Control Systems in PDF only on Docsity! Nonlinear Control Systems - Lecture 23 Local Asymptotic Stabilization Problem Consider
    with   having an equlibrium point at   and assume without loss of generality that   . We want to find a smooth state feedback law defined locally about  such that 1)    2)
  !  is locally asymptotically stable. First let’s consider the linearization about   ,  " #$% &(')   " *+ ,-. where #/10325425687 659  and *:; ) . We can easily establish the following theorem, Theorem: Suppose the linear approximation is asymptoti- cally stabilizable. Then any linear feedback that asymptot- ically stabilizes the linearization also asymptoticaly stabi- lizes (locally) the nonlinear system. Proof: Suppose the linear approximation is asymptotically stabilizable. Let < be a matrix such that #= ,* < is stable and set > <  . The corresponding closed loop nonlinear system is
    ?  <  @#& A* < B A '   - <  which has a linear approximation whose eigenvalues are in the LHS of the complex plane. So by Lyapunov’s indirect method, this system is locally asymptotically stable. C The related “instability” result is state and proven below. Theorem: If the linear apprxoimation is not controllable and there exists uncontrollable modes whose eigenvalues have positive real parts, then the original nonlinear system cannot be stabilized. Proof: Suppose the linear approximation has uncontrollable modes associated with eigenvalues having positive real parts. Let D be any smooth feedback law such that   . Then the linearization of the closed loop system is
  E3FG  ,8?HF  I 659   JK#& A*LE3F F  I 659 NM  Since  #POQ*R is uncontrollable, there exists a nonzero S such that TVUWYX # *[Z SD+ where U is the unstable eigenvalue and S is its associated eigenvector. So using the mode identified above, we see that\\)] SD J #& A*^F F  M S  U S which increases without bound since _a`  U cbd . So this is unstable regardless of our choice for  . C Note that these two theorems do not cover all possible cases since nothing is said if  #eOf*R has uncontrollable modes with zero real parts. It is this case that the notion of “zero dynamics” becomes useful. Again consider the system in normal form,
g -  g '
g '  gihjNjkj  jkjNj
gilNm-n gil
gilo p.q8OsrK ? Atq8OQr8 
r  u8 q8OsrK We again assume that q8OQr8 vw xOQ is an equilibrium point. We impose a feedback law of the form ytq8OsrK  X p( q8OsrK X{z 5g.- X,z -|gi' X jNjNj X{z l3m}-~gilk where z )O jkjNj O z lNm- are real numbers. This choice of control generates the following closed loop system
q  #q
r  u8 q8OsrK where #  €‚  y  jNjkj   y jNjkj ... ... ... . . . ...   jNjkj yX$z - X$z - X$z ' jNjkj X$z l3m}- ƒ…„„„„„† This matrix has the characteristic polynomial‡ ‰ˆi Š z  z - ˆ z ' ˆ ' jNjkj z l3m}- ˆ lNm- &ˆ l Obviously if we choose ‡ @ˆi to be a Hurwitz polynomial then we might expect this feedback law to stabilize the system. The following theorem provides conditions under which this conjecture holds true. Theorem: Suppose the equilibrium rL‹ of the zero dynamics is locally asymptotically stable and ‡Œ@ˆi is Hur- witz, then the preceding feedback law locally asymptotically stabilizes the nonlinear system. Docsity.com Proof: This is obvious given our earlier result son input-to- state stability of cascaded systems. Note that the matrix ŽE|F @u8q8OsrK Q F r I…N‘ ’3“ 9   ‘  “ characterizes the linear approximation of the zero dynamics at r=” . Our earlier stabilization theorem required  to have negative real parts. This preceding theorem is stronger for the zero dynamics can still be asymptotically stable even if its linearization is not stable. Note that we can stabilize systems whose linearization has a center eigen subspace provided we can choose an “output” function whose zero dynamics are asymptotically stable. In this case the locally asymptotically stabilizing cotnrol law would be y•Š–i• lNm-4˜— 0 X • l 4 — X{z  — X,z - • 4 — X jkjNj X,z lNm- • l3m}-4o— 7 where — is a special output map that we’ve chosen. Example: Consider the system
  €‚ -| ' X  h - -X h ' - = ' ƒ…„„† €‚ ™ ™  hy ƒ…„„† š  —  Šx› For this system F —F   T œ y Z• – —  " • 4 —  "  ' - ='F  • 4 — F   T ™ - y žAZ• – • 4 —  " ™  y = h Note that • – • 4 —  VŸ if  h Ÿ X y . So we can only find a normal form for points where hŸ y . In order to find the formal form, let’s consider the following local coordinate transformation,g - +  -  ¡ —  v ›g ' +  '  ¡ • 4 —  v '  ' -gih$+  h ¡ hg5›+ ›¢ ¡ - where   h and  › were freely chosen. The Jacobian of the local transformation isF}£F   €‚  œ y™ - y   y y œ ƒ…„„† which is nonsingular for all  . So the inverse transformation is -¤ g5› '˜ gi' X g '› h  g h ›  g - And the state equations for the system in normal form become,
g -  g '
g '  g › ™ g ›  g › @g ' X g '› X g h› } + ™ ™ gihN 
g h  X g h 
gk›˜ X ™ g h› =gi'3g5› Now the linear approximation at ¥¦ has the  #POQ*R matrices,#+§E~F F I 659   €‚ ž  y   ž X y  y   ƒ…„„† *¨d  © €‚  ™ y ƒ…„„† and this has exactly one uncontrollable mode corresponding to the eigenvalue U + . It’s zero dynamic can be shown to be stable. In particular if g.-agi'+ and we choose the control X g5› X{ª g ››™ ™ g h Then the resulting controlled system
g h  X g h X g5› X{ª g ››™ ™ gih
g5›˜ X ™ g h› can be easily shown to be asymptotically stable. So by our earlier theorem, the control y• – • 4 — TX • ' 4 — X{z  — X,z - • 4 — Z will stabilize the equilibrium  . If the output is not defined our if the zero dynamics are unstable, it may be possible to redesign the output function so our theorem can be used. Example: Consider the system
-¤  ' - = h'
'˜ 'v = Docsity.com