Chapter 3 Dynamic Behavior In this chapter we give a broad discussion of the behavior of dynamical sys- tems. We focus on systems modeled by di®erential equations, but consider the general nonlinear case. This allows us to discuss equilibrium points, stability, limit cycles and other key concepts of dynamical systems. We also introduce some measures of performance for input/output systems that provide additional tools for characterizing dynamic behavior. 3.1 Solving Di®erential Equations In the last chapter, we saw that one of the methods of modeling dynamical systems is through the use of ordinary di®erential equations (ODEs). A state space, input/output system has the form x_ = f(x; u) (3.1) y = g(x); where x Rn is the state, u Rp is the input, and y Rq is the output. The 2 2 2 smooth maps f : Rn Rp Rn and g : Rn Rq represent the dynamics £ ! ! and measurements for the system. We will focus in this text on single input, single output (SISO) systems, for which p = q = 1. We begin by investigating systems in which the input has been set to a function of the state, u = ®(x). This is one of the simplest types of feedback, in which the system regulates its own behavior. The di®erential equations in this case become x_ = f(x; ®(x)) = F (x): (3.2) As before, we write x = (x ; : : : ; x ) Rn for the state vector. 1 n 2 61 62 CHAPTER 3. DYNAMIC BEHAVIOR In order to understand the dynamic behavior of this system, we need to analyze the features of the solutions of this equation. While in some simple situations we can write down the solutions in analytical form, more often we must rely on computational approaches. We begin by describing the class of solutions for this problem. Initial Value Problems We say that x(t) is a solution of the di®erential equation (3.2) on the time interval t R to t R if 0 2 f 2 x_(t) = F (x(t)) for all t t < t . 0 · f A given di®erential equation may have many solutions. We will most often be interested in the initial value problem, where x(t) is prescribed at a given time t R and we wish to ¯nd a solution valid for all future time, t > t . 0 2 0 We say that x(t) is a solution of the di®erential equation (3.2) with initial value x Rn at t R if 0 2 0 2 x(t ) = x and x_(t) = F (x(t)) for all t t < t . 0 0 0 · f For most di®erential equations we will encounter, there is a unique solution that is de¯ned for t t t . The solution may de¯ned for all time t > t , 0 · · f 0 in which case we take t = . Because we will primarily be interested in f 1 solutions of the initial value problem for ODEs, we will often refer to this simply as the solution of an ODE. We will usually assume that t0 is equal to 0. In the case when F is indepen- Ä dent of time (as in equation (3.2)), we can do so without loss of generality by choosing a new independent (time) variable, ¿ = t t . ¡ 0 Example 3.1 (Damped oscillator). Consider a damped, linear oscillator, in- troduced in Example 2.3. The equations of motion for the system are mqÄ + bq_ + kq = 0; where q is the displacement of the oscillator from its rest position. We assume that b2 < 4km, corresponding to a lightly damped system (the reason for this particular choice will become clear later). We can rewrite this in state space form by setting x1 = q and x2 = q_, giving x_ 1 = x2 k b x_ = x x : 2 ¡m 1 ¡ m 2 3.1. SOLVING DIFFERENTIAL EQUATIONS 63 1 0.5 2 , x 0 1 x −0.5 −1 0 5 10 15 20 time (sec) Figure 3.1: Response of the damped oscillator to the initial condition x0 = (1; 0). In vector form, the right hand side can be written as x F (x) = 2 : k x b x ·¡ m 1 ¡ m 2¸ The solution to the initial value problem can be written in a number of di®erent ways and will be explored in more detail in Chapter 4. Here we simply assert that the solution can be written as bt x20 x (t) = e¡ 2m x cos !t + (x + ) sin !t 1 10 10 ! bt ³ b ´ b x20 x (t) = e¡ 2m x + x (! ) cos !t + x ! + (x + ) sin !t 2 20 10 ¡ 2 10 2 10 ! µ ¶ ¡ ¢ ¡ ¢ where x = (x ; x ) is the initial condition and ! = p4km b2=2m. This 0 10 20 ¡ solution can be veri¯ed by substituting it into the di®erential equation. We see that the solution is explicitly dependent on the initial condition and it can be shown that this solution is unique. A plot of the initial condition response is shown in Figure 3.1. We note that this form of the solution only holds for b2 4km < 0, corresponding to an \underdamped" oscillator. ¡ 64 CHAPTER 3. DYNAMIC BEHAVIOR Numerical Solutions One of the bene¯ts of the computer revolution that we can bene¯t from is that it is very easy to obtain a numerical solution of the di®erential equation when the initial condition is given. A nice consequence of this is as soon as we have a model in the form of (3.2), it is straightforward to generate the behavior of x for di®erent initial conditions, as we saw briey in the previous chapter. Modern computing environments allow simulation of di®erential equa- tions as a basic operation. In particular, MATLAB provides several tools for representing, simulating, and analyzing ordinary di®erential equations of the form in equation (3.2). To de¯ne an ODE in MATLAB, we de¯ne a function representing the right hand side of equation (3.2): function dxdt = sysname(t, x) dxdt = [ F1(x); F2(x); ... Fn(x); ]; Each function Fi(x) takes a (column) vector x and returns the ith element of the di®erential equation. The ¯rst argument to the function sysname, t, represents the current time and allows for the possibility of time-varying dif- ferential equations, in which the right hand side of the ODE in equation (3.2) depends explicitly on time. ODEs de¯ne in this fashion can be simulated by using the MATLAB ode45 command: ode45('file', [0,T], [x10, x20, ..., xn0]) The ¯rst argument is the name of the ¯le containing the ODE declaration, the second argument gives the time interval over which the simulation should be performed and the ¯nal argument gives the vector of initial conditions. The default action of the ode45 command is to plot the time response of each of the states of the system. Example 3.2 (Balance system). Consider the balance system given in Exam- ple 2.1 and reproduced Figure 3.2a. Suppose that a coworker has designed 3.1. SOLVING DIFFERENTIAL EQUATIONS 65 m 1.2 p theta 1 0.8 0.6 p 0.4 0.2 position (m), angle (deg) 0 F M −0.2 −0.4 0 2 4 6 8 10 12 14 16 18 20 time (sec) (a) (b) Figure 3.2: Balance system: (a) simpli¯ed diagram and (b) initial condition response. a control law that will hold the position of the system steady in the upright position at p = 0. The form of the control law is F = Kx; where x = (p; ; p_; _) R4 is the state of the system, F is the input, and 2 K = (k1; k2; k3; k4) is the vector of \gains" for the control law. The equations of motion for the system, in state space form, are p p_ d _ 2 3 = 2 1 3 dt p_ M + m ml cos ¡ bx_ + ml sin _2 + Kx 6_7 6 2 ¡ 7 6 7 6 J + ml ml cos mgl sin 7 4 5 4· ¸ · ¡ ¸5 p y = : · ¸ We use the following parameters for the system (corresponding roughly to a human being balanced on a stabilizing cart): M = 10 kg m = 80 kg b = 0.1 Ns/m J = 100 kg m2/s2 l = 1 m g = 9.8 m/s2 K = 1 120 4 20 ¡ ¡ This system can now be sim£ulated using MATLAB¤ or a similar numerical tool. The results are shown in Figure 3.2b, with initial condition x0 = (1; 0; 0; 0). We see from the plot that after an initial transient, the angle and position of the system return to zero (and remain there). 66 CHAPTER 3. DYNAMIC BEHAVIOR 100 100 80 80 60 60 x x 40 40 20 20 PSfrag replacements 0 PSfrag replacements 0 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 t t (a) (b) Figure 3.3: Solutions to the di®erential equation (3.3) (a) and (3.4) (b). Ä Existence and Uniqueness Without imposing some conditions on the function F the di®erential equa- tion (3.2) may not have a solution for all t, and there is no guarantee that the solution is unique. We illustrate this with two examples. Example 3.3 (Finite escape time). Let x R and consider the di®erential 2 equation dx = x2 (3.3) dt with initial condition x(0) = 1. By di®erentiation we can verify that the function 1 x(t) = (3.4) 1 t ¡ satis¯es the di®erential equation and it also satis¯es the initial condition.