chap13

Modern Control Engineering Chapter 13 LyapunovLyapunovStability Analysis Dr.Y. X. Liu Department of Automation Inner Mongolia University 2004-4-26Y. X. Liu, Inner Mongolia University2 Lecture Content LyapunovLyapunov Stability analysis Linear, Time-Invariant system Model-reference control system Quadratic optimal control 2004-4-26Y. X. Liu, Inner Mongolia University3 LyapunovStability Analysis For a given control system, stability is usually the most important thing to be determined. For linear and time invariant Routh’s stability criterion Nyquiststability criterion For nonlinear and/or time-varying The stability criteria above do not apply The second of LyapunovLyapunovdirect The second of Lyapunov Determination stability of linear, nonlinear, time invariant, and time-varying Solving quadratic optimal control problem 2004-4-26Y. X. Liu, Inner Mongolia University4 Lyapunovs In 1892 A. M. A. M. LyapunovLyapunovpresented. Determining the stability of dynamic system described by ordinary differential equations The first of Lyapunov Lyapunov Indirect Determining the stability of dynamic system depend on the of solutions of the ordinary differential equations. The Second of Lyapunov Lyapunov Direct Determine the stability of the system without solving the state equation. A.M.Lyapunov 2004-4-26Y. X. Liu, Inner Mongolia University5 The system defined by Equilibrium state A state xewhere fxe,t0 for all t. is called an equilibrium state. If the system is linear time invariant fx,tAx , then there exists only one equilibrium state if A is nonsingular, and there exist infinitely many equilibrium states if A is singular. For nonlinear system, there may be one or more equilibrium states, and correspond to the constant solution of the system xxefor all t. Any isolated equilibrium state can be shift to the origin of the coordinates, or f0,t0 for all t, by translation of coordinates. 00000 n ,t;t,tt; xxxvectornt vectorn t xxΦxΦ xf x xfx ⎩ ⎨ ⎧ − − 0 21 thussolution denote 1 .13 ,,,, , L 2004-4-26Y. X. Liu, Inner Mongolia University6 Stability in the sense of Lyapunov LyapunovStability 2004-4-26Y. X. Liu, Inner Mongolia University7 Definiteness Positive Definite Positive Semi-definite Negative Definite and Negative Semi-definite Indefiniteness Vx Both positive and negative values, no matter how small the region D is. 2004-4-26Y. X. Liu, Inner Mongolia University8 HermitianQuadratic Hermitian Here x is complex n-vector and P is a Hermitian matrix. For a real vector x and a real symmetric matrix P, Hermitian x*Px becomes equal to Quadratic xTPX. [] ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ nnnn n n x x pp pp xxPxVM L MM L L 1 1 111 1 * xx 2004-4-26Y. X. Liu, Inner Mongolia University9 Sylvester’s criterion 000 1 111 2212 1211 11 nnn n pp pp pp pp p L MM L L Positive definite Vx x*Px be positive definite is that all the successive principal minors of P be positive, that is Positive Semi-definite P is singular and all the principal minors are nonnegative. Negative definite the successive principal minors of P be negative and positive in turn. negative Semi-definite 2004-4-26Y. X. Liu, Inner Mongolia University10 LyapunovDirect If x 0 is an equilibrium for the system and there exists a continuously differentiable function, Vx, which contains the equilibrium x0 is stable x0 is asymptotically stable If x 0 is asymptotically stable and, in addition, V is radially unbounded then the x is globally asymptotically stable 2004-4-26Y. X. Liu, Inner Mongolia University11 Krasovskii’smathod xFxFxFdefine x f x f x f x f x f x f x f x f x f xx ff xF n nn n n n n ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ * 2 2 1 2 2 2 1 2 1 2 1 1 1 1 1 ˆ ,, ,, L L The system and assume that f00 and fx is differentiable with respect to xi,i1,2,,n. Define the Jacobian matrix is Hermitian symmetric if Fx is real. If is negative definite, then the equilibrium state x0 is asymptotically stable. xF ˆ xF ˆ largein stateally asymptotic is0 function Lyapunov * ⇒∞→∞→ xxasxfxf xfxfxV * 2004-4-26Y. X. Liu, Inner Mongolia University12 Lyapunovstability analysis of linear time-invariant system For system , a necessary and sufficient condition that the equilibrium state x0 be asymptotically stable in the large is that, given any positive-definite Hermitian or real symmetric matrix Q, there exits a positive- definite Hermitian or real symmetric matrix P such that Lyapunovfunction for this system is x*Px. If does not vanish identically along any trajectory, then Q may be choose semidefinite, and Q satisfiers xxA QPAPA− * xx QxV * − n AQ AQ Q rank n ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ −1 2 1 2 1 2 1 ⎦⎣ 2004-4-26Y. X. Liu, Inner Mongolia University13 Model-reference control system 1.Is a negative-definite matrix. 2.The control vector u can be chosen to make the scalar quantity M nonpositive BvtuAPM MPAPAxVPVLyapunov BvtuAA BvA tu d dd − −− − ,, 2 ,,define referenceModel ,,equationPlant * *** xfxe eeeee xfxeexxe xx xfx QPAPA− * x uxdCommand v Plant Model-reference system controller 2004-4-26Y. X. Liu, Inner Mongolia University14 Parameter-optimization System, where all eigenvalues of A have negative real parts, or origin x0 is asymptotically stable. It is desired to minimize the following perance index Since X0 is the given initial condition and Q is also given. P is a function of the elements of A. Hence, to minimize the perance index Jwill result in the optimal value of the adjustable parameter. xxA 000 * *** 0 * xx xxxx xx PJthereforexfor QPAPAobtainP dt d Qassume dtQJ →∞ −− ∫ ∞ 2004-4-26Y. X. Liu, Inner Mongolia University15 Quadratic optimal control BuA x x -K xu Given the system equations, determine the matrix K of the optimal control vector u, so as to minimize the perance index J. PBRK QPBPBRPAPA dtPJ RKKQBKAPPBKAP dt d RKKQ dtRKKQJBKA dtRuuQJKuBuA *1 *1* 0 * * * **** 0 ** 0 ** matrix optimal 0quation Riccati 000so stable, isA since set question − − ∞ ∞ ∞ − →∞ −−−− − − ∫ ∫ ∫ xxx xxxx xxxx xxxxx 2004-4-26Y. X. Liu, Inner Mongolia University16 References http//www.control.lth.se/people/personal/rjd ir/RiceUniversity/RiceLecture5-RJ.pdf http//web.nps.navy.mil/me/blackboard/me 4811/docs/notes5.pdf http//academic.csuohio.edu/simond/linearsys tems/stability/ http//www.aoe.vt.edu/cwoolsey/courses/A OE5984/Lectures/ http//www.ee.usyd.edu.au/tutorials_online/ matlab/PID/PID.htmlp Modern Control Engineering Chapter 13 LyapunovLyapunovStability Analysis Dr.Y. X. Liu Department of Automation Inner Mongolia University