TTK4150 Nonlinear Control Systems Department of ... - itslearning

TTK4150 Nonlinear Control SystemsDepartment of Engineering CyberneticsNorwegian University of Science and TechnologyFall 2009 - Assignment 1Due date: Wednesday 2 September at 16.00.1. (a) Calculate the vector norms ‖·‖ 1, ‖·‖ 2and ‖·‖ ∞for the following vectors:⎡ ⎤[ ] 1−1x 1 = , x7 2 = ⎣−9⎦3(Hint: See Appendix A in Khalil)(b) For a 2D vector, is ‖·‖ 2> ‖·‖ 1possible?Characterize all 2D vectors for which the ‖·‖ 2norm is equal to the ‖·‖ 1norm. Explaingraphically.2. Let⎡−2 1⎤0⎡ ⎤1A = ⎣ 1 −2 1 ⎦ , x = ⎣9⎦0 1 −2 3(a) Calculate the matrix norms ‖·‖ 1, ‖·‖ 2and ‖·‖ ∞for matrix A.(b) The following statement is true for any norm:‖Ax‖ ≤ ‖A‖ ‖x‖Show that it is true for the given matrix A and vector x, using the ‖·‖ 2norm.3. For any x, y ∈ R 2 define an inner product〈x, y〉 x T y.Show that|〈x, y〉| ≤ ‖x‖ 2‖y‖ 2.1

4. The following exercise is related to the computer game Duckmaze which will also beincluded to the other sets of exercise in the future. The present and future exercisesaim at analysing a nonlinear system and designing several controllers for the system.The controllers can be tested on a simulator in PIDstop (see http://www.pidstop.com/index.php?r_id=215) where a game based on this system is also available. How toupload the controller is described in the tutorial ‘How to upload a user-defined controller’available from the web page. The system we consider is a nonlinear mass-damper-springsystem (see Figure 1) with the following notations.x 1x 2x 3x 1dPosition of the box. This is zero when the spring is in its natural shape.Velocity∫of the box.(x1 − x 1d )dtThe desired position.F s Spring force (F s = f 3 x 3 1 + f 1 x 1 ).F d Damping force (F d = dẋ 1 ).f 1 Spring constant (linear effect).f 3 Spring constant (nonlinear effect).d Damping coefficient.m Mass of the box.g Gravity constant.w A disturbing force.u Input force to the system.and w is assumed to be unknown and will not be considered in the development of thesystem equations. By using Newton’s second law of motion the following mathematicalmodel is obtainedẋ 1 = x 2 (1)ẋ 2 = − f 3m x3 1 − f 1m x 1 − d m x 2 − g + u (2)mThis model is nonlinear. In this course you will learn how to analyze such nonlinearsystems and design controllers. However, with your current knowledge of linear controlsystems you are already able to design a simple controller by linearizing the system. Thatis the main purpose of this exercise. It will also allow you to recall important materialfrom the courses TTK4105 Control Engineering and TTK4115 Linear System Theory.(a) Linearize the system (1)-(2) about the point (x ∗ 1, x ∗ 2) = (0, 0). If you do not rememberhow to linearize, you should recover this from the book of the courses TTK4105Control Engineering and TTK4115 Linear System Theory.(b) Show that the system is controllable. What does this mean in connection with poleplacement?(c) Augment the linear model to include integral action. Call the new state x 3 . Explainwhat is achieved by doing this.(d) Design a linear state feedback controller u = −k 1 x 1 − k 2 x 2 − k 3 x 3 that places thepoles of the system inp 1 = −1, p 2 = −2, p 3 = −3(Hint: You may use the Matlab command place)k 1 , k 2 , and k 3 are the controller gains. Use the following numerical values: f 1 = 1,f 3 = 1, d = 1, g = 9.81, m = 1. Test it on the simulator. (If you want to make itthrough all skill levels, you need to place the poles differently.)2

(e) The solution of the augmented model with the gains calculated in the previousquestion is (the initial conditions of the states have been set to zero)Use this equation to show thatx 1 (t) = (1 − e −3t + 3e −2t − 3e −t )x 1d (3)|x 1 (t) − x 1d | ≤ ke −λt (4)where k and λ are constants greater than zero. Find k and λ.(f) Use the simulator and set the reference x 1d = 2 in the parameter window. Start thesimulator and save the data from the plot x 1 (t). Import these data into Matlab andplot |x 1 (t) − x 1d | against (4). Comment.Figure 1: Sketch of the mass-damper-spring system (as indicated by x ′ the system position ismeasured at the top of the mass).3

5. Letf(x) ={ 0√ for x < 0x else(a) Is f(x) globally Lipschitz? Explain.(b) Find the area for which f(x) is locally Lipschitz.6. For the following systems, find whether(1) f is locally Lipschitz in x on the whole state space(2) f is globally LipschitzBased on this information, say for which systems we can conclude about local or globalexistence and uniqueness of solutions.The systems are:(a) The pendulum equation with friction and constant input torque (see Section 1.2.1in Khalil)(b) The mass-spring equation with linear spring, linear viscous damping, Coulomb frictionand no external force (see Section 1.2.3 in Khalil)(c) The Van der Pol oscillator (Example 2.6 in Khalil)7. Exercise 1.4 in Khalil8. In some cases, nonlinear systems ẋ = f(x) with a complicated f(x) can be written in asimpler form by changing the coordinate vector x into a new coordinate vector z = ψ(x),where ψ is a continuously differentiable function (such that ψ −1 (z) exists and is alsocontinuously differentiable).With the new coordinate vector the corresponding system equations can be obtained byż = ∂ψ∣∂ψ ∣∣∣x=ψ(x)ẋ =∂x ∂x (x)f(x) = ∂ψ−1 (z)∂x (ψ−1 (z))f(ψ −1 (z))Although this expression may look more complicated, in some cases the resulting systemcan be simpler and more suitable for analysis than the original system, as illustrated inthe following exercise.Consider the nonlinear systemẋ 1 = x 2 + αx 1(β 2 − x 2 1 − x 2 2ẋ 2 = −x 1 + αx 2(β 2 − x 2 1 − x 2 2where α, β ≥ 0 are constants.Show that using polar coordinates (x 1 = r cos θ, x 2 = r sin θ) we can rewrite the originalsystem in terms of the new coordinates, to get a simpler equation set:ṙ = αr ( β 2 − r 2)˙θ = −1))4

9. Using (nonlinear) differential equations we can not only describe and analyze technicalsystems, but also biological, economical and other systems.A simplified representation of populations of foxes and rabbits is given by the followingequation set:wherex : Number of rabbitsẋ = x(α − βy)ẏ = −y(γ − δx)αx : Natural reproduction of rabbitsβxy : Rate of predationy : Number of foxesγy : Natural death of foxesδxy : Growth rate for foxesα, β, γ, δ : Parameters representing the interaction of the two speciesWe know that x ≥ 0, y ≥ 0, since the number of animals can not be negative.Note that the rate of predation is quite similar to the growth rate for foxes, but theycontain different parameters (as the fox population growth is not necessarily equal to therate at which it consumes the rabbits).(a) If initially x > 0, y = 0, what would happen to the rabbit population?(b) If initially x = 0, y > 0, what would happen to the fox population?(c) Find the equilibrium points of the system, and determine the type of each equilibriumpoint when all parameters are positive.(d) Is the existence of several isolated equilibrium points possible for linear systems?Why/why not?(e) Construct the phase portait for x ≥ 0, y ≥ 0 and discuss the qualitative behavior ofthe system. Choose the values α = 63, β = 5, γ = 457, δ = 6. Use the Matlab phaseportrait program pplane from http://math.rice.edu/~dfield/.(f) If initially x > 0 and y > 0, is it possible to arrive at x < 0 or y < 0? Explain youranswer.(g) Using Bendixson’s criterion (Lemma 2.2), specify a simply connected region in whichno periodic orbits are lying entirely. You may describe the region using inequalities.(See definiton in Khalil at p. 67.)5

10. Consider the nonlinear systemẋ 1 = ax 1 − x 1 x 2ẋ 2 = bx 2 1 − cx 2where a > 0, b > 0, c > 0.(a) Find the Jacobian of the right-hand side of the system.(b) Find all equilibrium points of the system. Determine the type of each isolatedequilibrium point for all values of a > 0, b > 0, c > 0.(c) For each of the following cases, construct the phase portait and discuss the qualitativebehavior of the system.1. a = 1, b = 1, c = 42. a = 1, b = 1, c = 10Do the phase portraits confirm the placement and type of equilibrium points whichyou found in (b)?6