Lecture Notes on Nonlinear Systems and Control

1 Lecture Notes on Nonlinear Systems and ControlSpring Semester 2018 ETH ZurichPeter Al Hokayem and Eduardo GallesteyABB Switzerland, 1 KCH-5405, Baden-D Analysis61 Main Concepts .. Models and Nonlinear Phenomena .. Typical Nonlinearities .. Nonlinearities .. with Memory .. Examples of Nonlinear Systems .. Reactor .. Second Order Systems .. Behavior of 2ndOrder Systems Near Equi-librium Points .. Cycles and Bifurcations .. Exercises .. 312 Ordinary Differential and Uniqueness .. and Differentiability .. and Sensitivity Equation .. Exercises .. 453 Lyapunov Stability Introduction .. Stability of Autonomous Systems .. s Direct Method .. s Indirect Method .. 531 Nonlinear Systems and Control Spring The Invariance Principle .. Instability Theorem.

2 Comparison Functions .. Stability of Nonautonomous Systems .. Existence of Lyapunov Functions .. Input-to-State Stability .. Stability of Discrete-Time Systems .. Exercises .. 744 Dissipative Introduction .. Dissipative Systems .. Passive Systems .. of Passivity for Linear Systems .. of Passive Systems .. Control .. Exercises .. 93II Control Design Techniques955 Feedback Control Introduction .. Linearization .. Integral Control .. Gain Scheduling .. Exercises .. 1086 Feedback Introduction .. Input-Output Linearization .. to Linear Systems .. Full State Feedback Linearization .. Stability .. Robustness .. Exercises .. 125P. Al Hokayem & E. Gallestey2 of 214 Nonlinear Systems and Control Spring 20187 Sliding Mode Introduction.

3 Sliding Mode Control .. Tracking .. Exercises .. 1358 Optimal Unconstrained Finite-Dimensional Optimization .. Finite-Dimensional Optimization .. Pontryagin s Minimum Principle .. Sufficient Conditions for Optimality .. Exercises .. 1519 State-Dependent Riccati State-Independent Riccati Equation Method .. on Control Design for LTI Systems .. State-Dependent Riccati Equation Approach .. Control for Nonlinear Systems .. Exercises .. 16510 Nonlinear Model Predictive Introduction .. Main Theoretical Elements .. Case Study: Tank Level Reference Tracking .. Chronology of Model Predictive Control .. Stability of Model Predictive Controllers .. Exercises .. 17911 State Estimation and Least Squares Estimation of Constant Vectors .. Weighted Least Squares Estimator.

4 Propagation of the State Estimate and its Uncertainty .. Kalman Filter .. Extended Kalman Filter .. Moving Horizon Estimation .. High Gain Observers .. Exercises .. 197P. Al Hokayem & E. Gallestey3 of 214 Nonlinear Systems and Control Spring 2018 Sample Exam 1200 Sample Exam 2207 References214P. Al Hokayem & E. Gallestey4 of 214 Nonlinear Systems and Control Spring 2018 PrefaceThe objective of this course is to provide the students with an introductionto Nonlinear Systems and the various methods of controlling I of the course introduces the students to the notions of nonlinearitiesand the various ways of analyzing existence and uniqueness of solutions toordinary differential equations, as well as understanding various notions ofstability and their II of the course arms the students with a variety of Control methodsthat are suitable for Nonlinear Systems and is designed in such a way as toput the student in the position to deploy Nonlinear Control techniques in chapters are combined with exercises that are geared towards attainingbetter understanding of the pros and the cons of the different Al Hokayem & E.

5 Gallestey5 of 214 Part IAnalysis6 chapter Main ConceptsWhen engineers analyze and design Nonlinear dynamical Systems in elec-trical circuits, mechanical Systems , Control Systems , and other engineeringdisciplines, they need to be able to use a wide range of Nonlinear analysistools. Despite the fact that these tools have developed rapidly since the mid1990s, Nonlinear Control is still largely a tough this course, we will present basic results for the analysis of nonlinearsystems, emphasizing the differences to linear Systems , and we will introducethe most important Nonlinear feedback Control tools with the goal of givingan overview of the main possibilities available. Additionally, the lectures willaim to give the context on which each of these tools are to be used. contains an overview of the topics to be considered in this ResultsModeling and SimulationWell posednessODE TheoryBifurcationsDisturbance RejectionSensorsObserversStabilizationUn certaintyLyapunovTrackingNonlinearitiesF eedback LinearizationSliding ModeEconomic OptimizationControl EffortOptimal ControlConstraintsModel Predictive ControlTable : Course Content7 Nonlinear Systems and Control Spring Nonlinear Models and Nonlinear PhenomenaWe will deal with Systems of the form: x1=f1(x1,x2.)

6 ,xn,u1,u2,..,um) x2=f2(x1,x2,..,xn,u1,u2,..,um).. xn=fn(x1,x2,..,xn,u1,u2,..,um)wherex Rnandu , we will neglect the time-varying aspect. In the analysis phase,external inputs are also often neglected, leaving system x=f(x).( )Working with an unforced state equation does not necessarily mean that theinput to the system is zero. It rather means that the input has been specifiedas a given function of the stateu=u(x).Definition system is said to be autonomous or time invariant if thefunctionfdoes not depend explicitly ont, that is, x=f(x).Definition point xis called equilibrium point of x=f(x)ifx( ) = xfor some impliesx(t) = xfort .For an autonomous system the set of equilibrium points is equal to theset of real solutions of the equationf(x) = 0. x=x2: isolated equilibrium point x= sin(x): infinitely many equilibrium points x= sin(1/x): infinitely many equilibrium points in a finite regionLinear Systems satisfy the following 2 properties:1.

7 Homogeneity:f( x) = f(x), R2. Superposition:f(x+y) =f(x) +f(y), x,y RnP. Al Hokayem & E. Gallestey8 of 214 Nonlinear Systems and Control Spring 2018 For example, consider the system given by the linear differential equation: x=Ax+Bu( )wherex Rn,u Rm,A Rn n,B Rn the solution is given byx(t) = expAtx0+ t0expA(t )Bu( )d .( )Note that the expression forx(t) is linear in the initial conditionx0andin the Control functionu( ). Nonlinear Systems are those Systems that do notsatisfy these nice we move from linear to Nonlinear Systems , we shall face a more difficultsituation. The superposition principle no longer holds, and analysis toolsnecessarily involve more advanced mathematics. Most importantly, as thesuperposition principle does not hold, we cannot assume that an analysis ofthe behavior of the system either analytically or via simulation may bescaled up or down to tell us about the behavior at large or small must be checked first step when analyzing a Nonlinear system is usually to linearize itabout some nominal operating point and analyze the resulting linear , it is clear that linearization alone will not be sufficient.

8 We mustdevelop tools for the analysis of Nonlinear Systems . There are two basiclimitation of linearization. First, since linearization is an approximation inthe neighborhood of an operating point, it can only predict the local behaviorof the Nonlinear system in the vicinity of that point. Secondly, the dynamicsof a Nonlinear system are much richer than the dynamics of a linear are essentially Nonlinear phenomena that can take place only in thepresence of nonlinearity; hence they cannot be described or predicted bylinear models. The following are examples of Nonlinear phenomena: Finite escape time:The state of an unstable linear system can goto infinity as time approaches infinity. A Nonlinear system s state,however, can go to infinity in finite time. Multiple isolated equilibrium points:A linear system can haveonly one equilibrium point, and thus only one steady-state operatingpoint that attracts or repels the state of the system irrespective of theinitial state.

9 A Nonlinear system can have more than one Al Hokayem & E. Gallestey9 of 214 Nonlinear Systems and Control Spring 2018 Limit cycles:A linear system can have a stable oscillation if it has apair of eigenvalues on the imaginary axis. The amplitude of the oscilla-tion will then depend on the initial conditions. A Nonlinear system canexhibit an oscillation of fixed amplitude and frequency which appearsindependently of the initial conditions. Chaos:A Nonlinear system can have a more complicated steady-statebehavior that is not equilibrium or periodic oscillation. Some of thesechaotic motions exhibit randomness, despite the deterministic natureof the system. Multiple Modes of behaviour:A Nonlinear system may exhibitvery different forms of behaviour depending on external parameter val-ues, or may jump from one form of behaviour to another behaviours cannot be observed in linear Systems , where the com-plete system behaviour is characterized by the eigenvalues of the systemmatrix Typical NonlinearitiesIn the following subsections, various nonlinearities which commonly occur inpractice are Memoryless NonlinearitiesMost commonly found nonlinearities are: Relay, see Relays appear when modelling mode changes.

10 Saturation, see Figure Saturations appear when modelling vari-ables with hard limits, for instance actuators. Dead Zone, see Figure Dead Zone appear in connection to actu-ator or sensor sensitivity. Quantization, see Figure Quantization is used to model discretevalued variables, often Al Hokayem & E. Gallestey10 of 214 Nonlinear Systems and Control Spring 2018uy 11 Figure : Relayuy kFigure : SaturationThis family of nonlinearities are called memoryless, zero memory or staticbecause the output of the nonlinearity at any instant of time is determineduniquely by its input at that time instant; it does not depend on the historyof the Nonlinearities with MemoryQuite frequently though, we encounter Nonlinear elements whose input-outputcharacteristics have memory; that is, the output at any instant of time maydepend on the recent or even entire, history of the the case of hysteresis one is confronted with a situation where thepath forward is not the same as the path backward.