MCE693/793: Analysis and Control of Nonlinear Systems

MCE693/793: Analysis and Control of Nonlinear Systems Systems of Differential Equations Phase Plane Analysis Hanz Richter Mechanical Engineering Department Cleveland State University

Systems of Nonlinear Differential Equations We often work with systems in the general form: ẋ = f(t,x) where solutions x(t) take values in R n and f(t,x) is a function defined on a subset of R R n. Whether the above systems admits one, none or multiple solutions given an initial point x(t 0 ) = x 0, is a fundamental question. Can we find a solution to How about ẋ = sign (x)? ẋ = x 1/3? (from Khalil, Sect. 2.2). We need a few mathematical concepts to answer existence and uniqueness of solutions. 2 / 36

Norms in Euclidean Space The n dimensional Euclidean space (over the real field) is formed by all vectors x = [x 1,x 2,...x n ] T where x i R for all i = 1,2...n. The norm x of a vector is a real-valued function satisfying four axioms for all x,y R n : 1. x 0 2. x = 0 x = 0 3. αx = α x for all α R 4. x+y x + y We are familiar with the Euclidean norm since childhood: x = n i=1 x2 i. But other norms are commonly used. Show that x = max i { x i } is a valid norm. How to extend the idea of norm to matrices will be discussed later 3 / 36

Continuity and Uniform Continuity of Functions We consider functions in Euclidean space, f : R n R m. We look at two equivalent definitions: 1. f is continuous at a point x if the sequence f(x k ) converges to f(x) for any sequence x k which converges to x. Succintly, f(x k ) x as x k x. 2. Given any ǫ > 0, we can find δ(ǫ,x) > 0 such that x y < δ(x,ǫ) f(x) f(y) < ǫ When δ(x,ǫ) is actually independent of x in the second definition, we speak of uniform continuity. Important: Uniform continuity obviously implies continuity, but the reverse is also true when f : S R m, where S R n is compact (closed and bounded). 4 / 36

Continuity in Some Variables and Piecewise Continuity The function f(x,y) = x sign (y) yx 2 is continuous in x but not in y. Specifically, continuity in a subset of variables is evaluated by regarding the remaining variables as constants. The function round(x) for x R is not continuous, but it is piecewise-continuous. That is, its domain can be partitioned into regions where the function is continuous. In this case, round(x) is continuous on any interval of the form: (k+0.5,k+1.5), k = 0,±1,±2,±3... 5 / 36

Lipschitz Condition, Local Existence and Uniqueness Theorem (Khalil 2.2) Let f(t,x) be piecewise continuous in t and suppose the Lipschitz condition f(t,x) f(t,y) L x y holds for some constant L > 0, for all x,y in some closed ball of radius r centered at x 0 and for all t in [t,t 0 ]. Then the differential equation ẋ = f(t,x) with initial condition x(t 0 ) = x 0 has a unique solution in an interval [t 0,t 0 +δ], for some δ > 0. Example: for f(x) = x 2, use the above Theorem to analyze existence and uniqueness. Can the solution be continued indefinitely for any initial conditions? 6 / 36

Global Existence and Uniqueness The previous example shows that solutions may be short-lived unless we place additional conditions. The Lipschitz condition will have to hold not only locally, but globally in x and in an extended set for t. Boundedness of f(t,x 0 ) will also be required. Theorem (Khalil 2.3) Let f(t,x) be piecewise continuous in t and suppose the conditions f(t,x) f(t,y) L x y f(t,x 0 ) h hold for all x,y R n, all t [t 0,t 1 ] and some h > 0. Then the differential equation ẋ = f(t,x) with initial condition x(t 0 ) = x 0 has a unique solution in the interval [t 0,t 1 ] 7 / 36

Constant Solutions (Equilibrium Points) Equilibrium points are solutions of the form x(t) = c, for some constant c, valid for all t. They reflect operating setpoints in engineering systems and must be studied carefully. Nonlinear systems may have empty, finite or infinite equilibrium sets. The last class may consist of a continuum of equilibria or countless isolated points, or a mix. Finding and characterizing the cardinality of equilibrium sets involves the solution of the nonlinear equations f(t,x) = 0 We point out that most of the time we use autonomous systems (where t does not appear in f), but it is possible to find equilibria in non-autonomous systems. For example: ẋ = f(t,x) = cos(t)x has x = 0 as the single equilibrium point, regardless of t. But the behavior of the solution away from zero is influenced by the cos(t) factor. 8 / 36

Examples: Equilibrium Finding Whenever an analytical solution to f(x) = 0 is not possible, we must use numerical tools. Matlab s fsolve is useful. Find all equilibrium points for the tunnel diode example in Khalil (1.2) and verify the solution provided there using fsolve. Find all equilibrium points for the second-order system ẋ = round(y) ẏ = sin(x) and sketch them in the phase plane (x-y plane) 9 / 36

Control Systems and Equilibrium A control system (a system with an exogenous input) includes an additional argument in f: ẋ = f(t,x,u) where u may be specified as a function u(t,x) with values in R m. Although is may be possible to find constant solutions when u is non constant (take f(x,u) = cos(u)x, then x = 0 is an equilibrium point regardless of how u varies), the accepted definition of equilibrium points for control systems involves setting u to some constant u 0. It is clear then that the equilibrium solutions depend on u 0. For equilibrium-finding purposes, u 0 is simply treated as a parameter of f. 10 / 36

Equilibrium Point Stability We know that equilibrium solutions are stationary, that is, the system remains at its initial condition x 0 indefinitely whenever x 0 is an equilibrium point. More importantly, we want to determine whether the system has the tendency to return tox 0 if a small initial condition offset is used, or if it moves away fromx 0. Stable Unstable 11 / 36

Equilibrium Point Stability In engineering we normally prefer systems that regulate themselves back to their original operating point when perturbed. When systems don t have that natural tendency (example: magnetic levitation), we must add feedback controls to change system behavior from unstable to stable. A precise definition of stability is pending. With Lyapunov s linearization method, we determine the stability of an equilibrium point of a nonlinear system by examining the stability of the linearized system about the origin. Except for one case (zero eigenvalues), the stability of the linearized system is the same as the local stability of the nonlinear system. 12 / 36

Review of Small-Signal (Jacobian) Linearization Let x 0 be an equilibrium point for ẋ = f(x,u) for u = u 0, and consider f sufficiently smooth to possess a Taylor series expansion at (x 0,u 0 ): f(x,u) f(x 0,u 0 )+A(x x 0 )+B(u u 0 )+H.O.T. where A = J x (x 0,u 0 ) and B = J u (x 0,u 0 ) are the Jacobians of f(x,u) relative to x and u evaluated at the equilibrium point. Define δ x = x x 0 and δ u = u u 0. Neglecting the higher-order terms we obtain a linear dynamic description of δ x with δ u as an input: δ x = Aδ x +Bδ u Note that the above is a good approximation provided: 1. δ x and δ u are small (solutions remain close to equilibrium values). Othewise the H.O.T. introduce significant differences. 2. f(x 0,u 0 ) = 0, that is, we linearize relative to an equilibrium point. It is a popular mistake to linearize at non-equilibrium points (trajectories) using the above formulas. Linearization methods relative to trajectories are available. 13 / 36

Equilibria in Linear Systems Consider the n-state linear system ẋ = Ax As we saw before, the equilibrium set is either {0} (when A is nonsingular) or a linear subspace of R n (when A is singular). We also know that the system is asymptotically stable whenever all eigenvalues of A have negative real parts. This implies non-singularity and uniqueness of equilibrium. If at least one eigenvalue is zero, singularity follows and a dense equilibrium set arises. Second-order systems are highly relevant and allow graphical study through phase-plane analysis. In linear systems, we can completely characterize solution behavior by examining the eigenvalues and eigenvectors of A. In the second-order case, graphical representations are very useful. 14 / 36

2nd-Order Linear Systems: Distinct real eigenvalues When the two eigenvalues are real, distinct and nonzero, two linearly-independent real eigenvectors v 1 and v 2 can be found. Then A is diagonalizable by the transformation x = Tz, with T = [v 1 v 2 ]. In z (modal) coordinates, the system becomes decoupled: ż 1 = λ 1 z 1 ż 2 = λ 2 z 2 The solutions are single exponential functions, and satisfy z 2 = cz λ 2/λ 1 1 for some constant c dependent on initial conditions. The signs of λ 1 and λ 2 determine stability and the shape of the phase portrait. 15 / 36

Stable and Unstable Nodes in Modal Phase Space If both eigenvalues are negative, we have a stable node. If at least one is positive, we have an unstable node. In modal space, eigenvectors are (0,1) and (1,0). Recall that an initial condition belonging to the span of an eigenvector generates a trajectory along the eigenvector. 16 / 36

Stable and Unstable Nodes in Original Phase Space In phase space, eigenvectors are v 1 (slow) and v 2 (fast). All solutions are linear combinations of eigensolutions. Again, an initial condition belonging to the span of an eigenvector generates a trajectory along the eigenvector. 17 / 36

Saddle Point If the eigenvalues have opposite signs, the equilibrium point is a saddle. Examine solution behaviors near the stable and unstable eigenvectors. 18 / 36

Straight-Line Solutions If the eigenvalues are real and one of them is zero, there are straight line solutions. To see this, note that there is still a set of linearly-independent eigenvectors which diagonalize the system via x = Mz, with M = [v 1 v 2 ]. In these modal coordinates: ż 1 = 0 ż 2 = λ 2 z 2 The fact that phase trajectories are straight lines is obvious from these equations. Note that A is singular, so the equilibrium set is dense (the null space of A). 19 / 36

Straight-Line Solutions... If the eigenvalues are real and both of them are zero, A could be the zero matrix. In this case, there are no trajectories and the entire phase plane is made up of equilibrium points. But there are matrices whose nullspace has dimension 1 (nonzero A with λ 1 = λ 2 = 0). In this case, diagonalization leads to ż 1 = z 2 ż 2 = 0 Again, trajectories in modal space are horizontal lines. The direction of the trajectories depends on the initial condition for z 2. 20 / 36

Complex Eigenvalues: Stable Focus Points When the eigenvalues have the form a±bi with a < 0 and b 0, trajectories spiral into the equilibrium point. Clocwkise or counterclockwise? 21 / 36

Complex Eigenvalues: Focus Points When the eigenvalues have the form a±bi with a > 0 and b 0, trajectories spiral out of the equilibrium point. Clockwise or counterclockwise? 22 / 36

Imaginary Axis Eigenvalue(s): Center Points When the eigenvalues have the form bi with b 0, trajectories are periodic. The phase plane is partitioned by periodic orbits, that is, any two periodic orbits have empty intersection and the union of all periodic orbits is the entire plane. We will not call these orbits limit cycles, because there are no initial conditions outside an orbit which converge to it. 23 / 36

Nonlinear 2nd-Order Systems - Isoclines Method For nonlinear second-order systems, phase portraits can be generated by: 1. Computer simulation and graphics 2. Analytical solution and elimination of time from solutions 3. Method of isoclines 4. Other methods Methods 2 and 3 can be used to obtain insightful sketches and induce learning. We will focus on the method of isoclines. Given ẋ 1 = f 1 (x 1,x 2 ) ẋ 2 = f 2 (x 1,x 2 ) we can show that the slope of phase trajectories is given by dx 2 dx 1 = f 2(x 1,x 2 ) f 1 (x 1,x 2 ) 24 / 36

Isoclines Method To use the isoclines method, find and plot all equilibrium points. Then determine the locus of phase space points having particular values of slope that might be informative, such as,±1 and 0. From dx 2 = f 2(x 1,x 2 ) dx 1 f 1 (x 1,x 2 ) note that the slope at equilibrium points is undefined. Example: Let f 1 (x 1,x 2 ) = x 2 and f 2 (x 1,x 2 ) = x 1 +x 2. Sketch the phase portrait. Verify against known characteristics of linear systems. 25 / 36

Analytical Method In some cases, explicit solutions x 1 (t) and x 2 (t) can be found, given an initial point (x 10,x 20 ). If t can be solved for as a function of x 1 (or x 2 ), direct substitution into x 2 (t) (or x 1 (t)) results in an implicit relationship describing the phase trajectory passing through (x 10,x 20 ). Example: Use the analytical method for the previous example. 26 / 36

Guidelines for Computer-Generated Phase Portraits 1. Take advantage of symmetry (also valid for hand sketching, see S&L 2.1.3.) 2. Negative-time solutions: start close to the equilibrium point and compute backwards. Define τ = t and see what happens to the system relative to this new time. See Khalil, 1.2.5 3. Use Matlab s ode45 or stiff methods as appropriate and compute solutions for short times. 4. See S&L 2.3 to estimate time spans needed for given trajectory lengths. 27 / 36

Limit Cycles - Background Definitions Consider the autonomous system ẋ = f(x) and let x = φ(x 0,t) be a solution with x(0) = x 0, assumed valid for t (, ). Define: The positive semiorbit through x 0 : γ + (x 0 ) = {φ(x 0,t) : 0 t < } The negative semiorbit through x 0 : γ (x 0 ) = {φ(x 0,t) : < t 0} 28 / 36

Limit Cycles - Background Definitions Definition of limit points and sets: If there is a sequence of times {t n } with t n as n such that the sequence {φ(t n,x)} p, we call p a positive limit point of the solution. If there is a sequence of times {t n } with t n as n such that the sequence {φ(t n,x)} p, we call p a negative limit point of the solution. The set of all positive limit points of φ(t,x 0 ) is the positive limit set of the solution. The set of all negative limit points of φ(t,x 0 ) is the negative limit set of the solution. 29 / 36

Limit Cycles - Background Definitions Definitions of periodic solutions and closed orbits: The solution φ(t,x 0 ) is nontrivial periodic if x 0 is not an equilibrium point and there is a constant T > 0 such that φ(t+t,x 0 ) = φ(x 0 ) for all t 0. The period is the smallest T that meets the above definition. If φ(t,x 0 ) is nontrivial periodic, the set {x R n : x = φ(t,x 0 ) for some t} is the periodic orbit or closed orbit associated with the solution. 30 / 36

Limit Cycles - Definition (Khalil 7.1) A limit cycle is a closed orbit γ such that γ is the positive limit set of some positive semiorbit γ + (x 0 ) or the negative limit set of some negative semiorbit γ (x 0 ), for some x 0 / γ. According to this definition, are linear harmonic oscillator solutions limit cycles? How about nonlinear pendulum oscillations? 31 / 36

The van der Pol Equation and its Limit Cycle The van der Pol equation ẍ µ(1 x 2 )ẋ+x = 0 was found by the eponymous Dutch researcher when investigating oscillations in circuits with vacuum tubes. We can interpret the equation as a mass-spring-damper system with nonlinear damping (which can have positive, negative or zero coefficient according to x). 3 van der Pol equation trajectories, µ=0.7 2 1 ẍ µ(1 x 2 )ẋ+x = 0 ẋ 0-1 -2-3 -4-3 -2-1 0 1 2 3 x 32 / 36

van der Pol Equation We use the van der Pol equation (µ 0) for an overview of equilibrium points and linearization.we also use it to demonstrate the isoclines method and computation of time from trajectories. 1. The origin is the only equilibrium point 2. The eigenvalues of the linearized system matrix A always have positive real parts for µ > 0. 3. Depending on µ, the origin may be a node or a focus, but always unstable for µ > 0. 4. Find values of µ to obtain each of the above 2 cases. 33 / 36

Limit Cycles in 2nd-Order Systems: 3 Theorems Knowing that the van der Pol equation is unstable at its equilibrium point, can we detect the presence of a limit cycle? We will consider a second-order system in the form of Eq. 2.1 in S&L: ẋ 1 = f 1 (x 1,x 2 ) ẋ 2 = f 2 (x 1,x 2 ) Poincaré s Theorem (index theorem): Let N be the total number of centers, foci and nodes enclosed by a limit cycle. Let S be the total number of enclosed saddle points. If a limit cycle exists, then N = S +1. The above implies that there can t be a limit cycle that doesn t contain an equilibrium point inside. The theorem provides a necessary condition for the existence of a limit cycle, not a sufficient condition. Make sure you have your logic straight. 34 / 36

Poincaré-Bendixson Theorem Statement from S&L: If a trajectory remains in a finite region of the phase plane for all t 0, then exactly one of the following holds: 1. The trajectory converges to an equilibrium point 2. The trajectory converges to a limit cycle 3. The trajectory is itself a limit cycle Examples 35 / 36

Poincaré-Bendixson and Bendixson Theorems Poincaré-Bendixson: Statement from Khalil: Let γ + be a bounded positive semiorbit and let L + be its positive limit set. If L + contains no equilibrium points, then it is a periodic orbit. Bendixson (S&L): If a limit cycle exists in a region Ω of the phase plane, then f 1 vanish and change sign somewhere in Ω. x 1 + f 2 x 2 must See the interesting proof with Stokes theorem in S&L. Examine the applicability of the three theorems to the van der Pol equation. Again, we have only a necessary condition. In general, limit cycle detection is a very hard problem. But when they don t exist, ruling them out is immediate using the above theorems. 36 / 36