Chapter 3Dynamic BehaviorIn this chapter we give a broad discussion of the behavior of dynamical sys-tems. We focus on systems modeled by differential equations, but considerthe general nonlinear case. This allows us to discuss equilibrium points,stability, limit cycles and other key concepts of dynamical systems. Wealso introduce some measures of performance for input/output systems thatprovide additional tools for characterizing dynamic behavior.3.1 Solving Differential EquationsIn the last chapter, we saw that one of the methods of modeling dynamicalsystems is through the use of ordinary differential equations (ODEs). Astate space, input/output system has the form˙x = f(x, u)y = g(x),(3.1)where x ∈ Rnis the state, u ∈ Rpis the input, and y ∈ Rqis the output. Thesmooth maps f : Rn× Rp→ Rnand g : Rn→ Rqrepresent the dynamicsand 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 afunction of the state, u = α(x). This is one of the simplest types of feedback,in which the system regulates its own behavior. The differential equationsin this case become˙x = f (x, α(x)) = F (x). (3.2)As before, we write x = (x1, . . . , xn) ∈ Rnfor the state vector.6162 CHAPTER 3. DYNAMIC BEHAVIORIn order to understand the dynamic behavior of this system, we need toanalyze the features of the solutions of this equation. While in some simplesituations we can write down the solutions in analytical form, more often wemust rely on computational approaches. We begin by describing the classof solutions for this problem.Initial Value ProblemsWe say that x(t) is a solution of the differential equation (3.2) on the timeinterval t0∈ R to tf∈ R if˙x(t) = F (x(t)) for all t0≤ t < tf.A given differential equation may have many solutions. We will most oftenbe interested in the initial value problem, where x(t) is prescribed at a giventime t0∈ R and we wish to find a solution valid for all future time, t > t0.We say that x(t) is a solution of the differential equation (3.2) with initialvalue x0∈ Rnat t0∈ R ifx(t0) = x0and ˙x(t) = F (x(t)) for all t0≤ t < tf.For most differential equations we will encounter, there is a unique solutionthat is defined for t0≤ t ≤ tf. The solution may defined for all time t > t0,in which case we take tf= ∞. Because we will primarily be interested insolutions of the initial value problem for ODEs, we will often refer to thissimply as the solution of an ODE.We will usually assume that t0is equal to 0. In the case when F is indepen-Ädent of time (as in equation (3.2)), we can do so without loss of generalityby choosing a new independent (time) variable, τ = t − t0.Example 3.1 (Damped oscillator). Consider a damped, linear oscillator, in-troduced in Example 2.3. The equations of motion for the system arem¨q + b ˙q + kq = 0,where q is the displacement of the oscillator from its rest position. Weassume that b2< 4km, corresponding to a lightly damped system (thereason for this particular choice will become clear later). We can rewritethis in state space form by setting x1= q and x2= ˙q, giving˙x1= x2˙x2= −kmx1−bmx2.3.1. SOLVING DIFFERENTIAL EQUATIONS 630 5 10 15 20−1−0.500.51time (sec)x1, x2Figure 3.1: Response of the damped oscillator to the initial condition x0=(1, 0).In vector form, the right hand side can be written asF (x) =·x2−kmx1−bmx2¸.The solution to the initial value problem can be written in a number ofdifferent ways and will be explored in more detail in Chapter 4. Here wesimply assert that the solution can be written asx1(t) = e−bt2m³x10cos ωt + (x10+x20ω) sin ωt´x2(t) = e−bt2mµ¡x20+ x10(ω −b2)¢cos ωt +¡x10ω +b2(x10+x20ω)¢sin ωt¶where x0= (x10, x20) is the initial condition and ω =√4km − b2/2m. Thissolution can be verified by substituting it into the differential equation. Wesee that the solution is explicitly dependent on the initial condition and itcan be shown that this solution is unique. A plot of the initial conditionresponse is shown in Figure 3.1. We note that this form of the solution onlyholds for b2− 4km < 0, corresponding to an “underdamped” oscillator.64 CHAPTER 3. DYNAMIC BEHAVIORNumerical SolutionsOne of the benefits of the computer revolution that we can benefit from isthat it is very easy to obtain a numerical solution of the differential equationwhen the initial condition is given. A nice consequence of this is as soon aswe have a model in the form of (3.2), it is straightforward to generate thebehavior of x for different initial conditions, as we saw briefly in the previouschapter.Modern computing environments allow simulation of differential equa-tions as a basic operation. In particular, MATLAB provides several toolsfor representing, simulating, and analyzing ordinary differential equationsof the form in equation (3.2). To define an ODE in MATLAB, we define afunction 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 elementof the differential equation. The first 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 define in this fashion can be simulated by using the MATLABode45 command:ode45(’file’, [0,T], [x10, x20, ..., xn0])The first argument is the name of the file containing the ODE declaration,the second argument gives the time interval over which the simulation shouldbe performed and the final argument gives the vector of initial conditions.The default action of the ode45 command is to plot the time response ofeach 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 designed3.1. SOLVING DIFFERENTIAL EQUATIONS 65pFMmθ0 2 4 6 8 10 12 14 16 18 20−0.4−0.200.20.40.60.811.2time (sec)position (m), angle (deg)p theta(a) (b)Figure 3.2: Balance system: (a) simplified diagram and (b) initial conditionresponse.a control law that will hold the position of the system steady in the uprightposition at p = 0. The form of the control law isF = Kx,where x = (p, θ, ˙p,˙θ) ∈ R4is the state of the system, F is the input,
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