Lecture 5Linear SystemsNonlinear SystemsNonlinear Systems (cont)Equilibrium Points for Linear SystemsNonlinear Example - Duffing EquationPendulum ExampleReducing Non-zero Equilibrium Points to the OriginOther Characteristics of Nonlinear SystemsNominal MotionExample Using the Duffing EquationLyapunov Stability TheoryStability Theory DefinitionsConcepts of Stability: Stability and InstabilityStability and Asymptotic StabilityRate of Convergence: Exponential StabilityLocal and Global StabilityStability Theory for Linear SystemsStability Theory for Linear SystemsLessons LearnedLinearization and Local StabilityLyapunov’s Linearization MethodInput-Output StabilityInput-Output StabilityInput-Output StabilityLecture 6Lyapunov Stability Theory: MotivationLyapunov’s Direct Method(2nd Theorem)Lyapunov’s Direct Method(Motivating Example)Lyapunov’s Direct Method(Motivating Example, continued)Lyapunov’s Direct MethodComputing VdotLyapunov’s Direct Method (cont’d)Lyapunov-Like LemmaLyapunov’s Theorem for Local StabilityUniform Ultimate Boundedness(UUB)UUB by Lyapunov ExtensionExample: UUB by Lyapunov ExtensionGlobal Stability and Radial UnboundednessExampleSelecting Lyapunov FunctionsSelecting Lyapunov FunctionsSelecting Lyapunov Functions (cont)RemarksExamplesExamplesExampleExample (cont)ExampleRegion of ASExampleRemarkLyapunov Analysis for LTI SystemsLTI ExampleLecture 5J.J. Slotine, W. Li, Applied Nonlinear Control, Prentice Hall, 1995.Khalil, H., K., Nonlinear Systems, Prentice Hall, 2002 Nonlinear Systems Reference Texts:Linear Systems() (),,xnxt Axt x R t R=∈∈• Unique equilibrium point if is nonsingular• Stable equilibrium if • System response is a composition of natural modes•IfA()()Re 0Aλ<,,,xunnxAxBuxRuRtR=+ ∈ ∈ ∈- Principle of superposition- If BIBO stability in presence of - Sinusoidal input produces sinusoidal output of same frequency()()Re 0Aλ<→uK.A. Wise2These same properties do not hold for nonlinear systemsNonlinear Systems(, ), ,xnxftx xR tR=∈∈General representation of nonlinear dynamicsThe solution is called state trajectory.00)(),( xtxtxx==Definition: The nonlinear system is called autonomous if the dynamics is not explicitly depending upon time:)(xfx=Definition: A state is called an equilibrium state (point) of the system, if once is equal to , it remains equal to for all future time. It can be found by solving the nonlinear algebraic equations:*x)(tx*x*x)(0*xf=K.A. Wise3Nonlinear Systems (cont)(, ), ,() (),(,),xunnxftx xR tRxfxgxuuRxfxu=∈∈=+ ∈=General representation of nonlinear dynamics• Multiple equilibrium points are possible• Stability depends upon initial conditions• Can exhibit limit cycles (self excited oscillations)• Exhibit bifurcations− Changes in parameters can cause stability and the number of equilibrium points to changeK.A. Wise4Equilibrium Points for Linear SystemsxAx=A linear time-invariant systemhas a single equilibrium point at the origin, if A is non-singular. If A is singular, it has an infinity of equilibrium points, which are contained in thenull-space of the matrix A, i.e. by the subspace defined by Ax=0. This implies that the equilibrium points are not isolated. Example:2221xxxx−==0=+ xxAll the points on the ,i.e. x axis are equilibrium points.02=xK.A. Wise5Nonlinear Example - Duffing Equation30xxxα++=Undamped()3**0xxα+=Solve for equilibriums*0, , ,xi iαα=−0α=Is a critical bifurcation valueK.A. Wise6*xCalled a pitchfork bifurcationαPendulum ExampleθMR0sin2=++θθθgMRbRMθθ==21, xxEquations of motion:State-space representation:122221sin xRgxMRbxxx−−==MgEquilibrium points:K.A. Wise7[][][]",02,0,00ππ0sin,012== xxReducing Non-zero Equilibrium Points to the OriginLet be the equilibrium point of interest for the dynamics:*x0)(),(*== xfxfxIntroduce new variable*xxy −=Notice, that*xyx +=0*=xThen, because )(*xyfyx +==Thus, will be the equilibrium point in the new variables. 0=yK.A. Wise8Other Characteristics of Nonlinear SystemsHopf BifurcationChanges in system parameters can cause limit cyclesChaosFor linear systems, small changes to initial conditions producessmall changes in the systems’ responseFor chaotic system, response is very sensitive to initial condition. This prevents one from predicting the system output.Chaotic systems are deterministic systems, not random motion.K.A. Wise9Nominal MotionLet be the solution of the equation:)(*tx0*)0(),( xxxfx ==Perturb the initial condition . Let be the corresponding solution.Define:00)0( xxxδ+=)(tx)()()(*txtxte −=Then0),0(,)0(),,())(())()(()(0**===−+= tgxetegtxftetxfteδK.A. Wise10)(*tx)(tx)(te1x2xThus, the error dynamics are non-autonomous.Example Using the Duffing Equation0321=++ xkxkxmConsider the following autonomous systemLet be the solution with the initial condition . )(*tx00*)( xtx =Perturb the initial condition . Let be the corresponding solution.Define:00)0( xxxδ+=)(tx)()()(*txtxte −=Then()0])(3)(3[2**2321=++++ txetxeekekemThus, we have non-autonomous error equation.It becomes non-autonomous due to the nonlinearity!K.A. Wise11Lyapunov Stability TheoryAlexander Michailovich Lyapunov1857-1918• Russian mathematician and engineer who laid out the foundation of the Stability Theory• Results published in 1892, Russia• Translated into French, 1907• Reprinted by Princeton University, 1947• American Control Engineering Community Interest, 1960’sK.A. Wise12Stability Theory DefinitionsEquilibrium State Stability()()()000exfx xt x fx===Nonlinear Dynamics:Equilibrium ConditionDefinition: Stable In the Sense of Lyapunov (SISL)()()00An equilibrium state is said to be stable ISL if, 0, a ,if , then .eeexRrRxx r xtx Rtt∀>∃ ∋−< −<∀>Definition: Asymptotically Stable (AS)()An equilibrium state is said to be AS if it is stable ISL, and lim .eetxxtx→∞=Definition: Asymptotically Stable in the Large (ASL)0An equilibrium state is said to be ASL, if it is AS and holds .exx∀K.A. Wise13Concepts of Stability: Stability andInstabilityDefinition: The equilibrium point is said to be stable if for all R > 0 there exists r(R) > 0, such that if , then for all .