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Nonlinear Systems: an Introduction to Lyapunov Stability Theory A. Pascoal, Jan. 2014

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Linear versus Nonlinear Control Nonlinear Plant Linear based control laws - Lack of global stability and performance results + Good engineering intuition for linear designs (local stability and performance) - Poor physical intuition Nonlinear control laws + Powerful robust stability analysis tools + Possible deep physical insight - Need for stronger theoretical background - Limited tools for performance analysis

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Nonlinear Control: Key Ingredients AUV speed control Dynamics Nonlinear Plant Objective: generate T(t) so that tracks the reference speed Tracking error Error Dynamics

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Nonlinear Control: Key Ingredients Error Dynamics Nonlinear Control Law

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Nonlinear Control: Key Ingredients Tracking error tends to zero exponentially fast. Simple and elegant! Catch: the nonlinear dynamics are known EXACTLY. Key idea: i) use “simple” concepts, ii) deal with robustness against parameter uncertainty. New tools are needed: LYAPUNOV theory

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Lyapunov theory of stability: a soft Intro (free mass, subjected to a simple motion resisting force) vfv v m/f 0v t v=0 is an equilibrium point; dv/dt=0 when v=0! v=0 is attractive (trajectories converge to 0) SIMPLE EXAMPLE

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Lyapunov theory of stability: a soft Introvfv0 v How can one prove that the trajectories go to the equilibrium point WITHOUT SOLVING the differential equation? (energy function) V positive and bounded below by zero; dV/dt negative implies convergence of V to 0!

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Lyapunov theory of stability: a soft Intro What are the BENEFITS of this seemingly strange approach to investigate convergence of the trajectories to an equilibrium point? V positive and bounded below by zero; dV/dt negative implies convergence of V to 0! vf(v) f a general dissipative force v 0 Q-I Q-III e.g. v|v| Very general form of nonlinear equation! vfv

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Lyapunov theory of stability: a soft Intro State vector Q-positive definite 2-D case

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1x22x11x1 V positive and bounded below by zero; dV/dt negative implies convergence of V to 0! x tends do 0!

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Lyapunov theory of stability: a soft Intro Shifting Is the origin always the TRUE origin? mg y y-measured from spring at rest Examine if yeq is “attractive”! Equilibrium point y: dx/dt=0 Equilibrium point y eq : dx/dt=0 Examine the ZERO eq. Point!

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Lyapunov theory of stability: a soft Intro Shifting Is the origin always the TRUE origin? Examine if x ref (t) is “attractive”! x ref (t) is a solution Examine the ZERO eq. Point!

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Lyapunov theory of stability: a soft Intro Control Action Nonlinearplant yu Static control law Investigate if 0 is attractive! is attractive!

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Lyapunov Theory Stability of the zero solution 0 x-space The zero solution is STABLE if

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Lyapunov Theory 0 x-space The zero solution is locally ATTRACTIVE if Attractiveness of the zero solution

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Lyapunov Theory The zero solution is locally ASYMPTOTICALLY STABLE if it is STABLE and ATTRACTIVE (the two conditions are required for Asymptotic Stability!) One may have attractiveness but NOT Stability!

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Key Ingredients for Nonlinear Control Lyapunov Theory (a formal approach)

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Lyapunov Theory (the two conditions are required for Asymptotic Stability!)

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Lyapunov Theory There are at least three ways of assessing the stability (of an equilibrium point of a) system: Solve the differential equation (brute-force)Solve the differential equation (brute-force) Linearize the dynamics and examine the behaviourLinearize the dynamics and examine the behaviour of the resulting linear system (local results for hyperbolic eq. points only) Use Lypaunov´s direct method (elegant and powerful,Use Lypaunov´s direct method (elegant and powerful, may yield global results)

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Lyapunov Theory

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If then the origin is globally asymptotically stable

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Lyapunov Theory What happens when Is the situation hopeless? No! Suppose the only trajectory of the system entirely contained in is the null trajectory. Then, the origin is asymptotically stable (Let M be the largest invariant set contained in . Then all solutions converge to M. If M is the origin, the results follows) Krazovskii-LaSalle

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y Lyapunov TheoryKrazovskii-La Salle

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y Lyapunov TheoryKrazovskii-La Salle f(.), k(.) – 1st and 3rd quadrants f(0)=k(0)=0 V(x)>0!

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y Lyapunov TheoryKrazovskii-La Salle Examine dynamics here! Trajectory leaves unless x 1 =0! M is the origin. The origin is asymptotically stable!

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Lyapunov TheoryKrazovskii-La Salle An application exampl (physically motivated control law) v-v|v| AUV moving in the water with speed v under the action of the applied force u. Objective: drive the position x of the AUV to x* (by proper choice of u) x 0 x*

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Lyapunov TheoryKrazovskii-La Salle An application example Suggested control law Control law exhibits Proporcional + Derivative actions The plant itself has a pure integrator (to drive the static error to 0)

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Lyapunov Theory Krazovskii-La Salle Show asymptotic stability of the (equilibrium point of the) system Step 1. Start by re-writing the equations in terms of the variables that must be driven to o. Objective:

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Lyapunov Theory Krazovskii-La Salle Show asymptotic stability of the (equilibrium point of the) system Step 2. Prove global asymptotic stability of the origin is an equilibrium point of the system! Seek inspiration from the spring-mass-dashpot system y

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Lyapunov Theory Krazovskii-La Salle Show asymptotic stability of the (equilibrium point of the) system V(x)>0! Using La Salle´s theorem it follows that the origin is globally asymptotically stable (GAS)

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