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2.0 Mathematical Backgroundsummarize

2.2 Solving the Differential Equation

T he goal in control systems design is to add a controller to the plant so that the resulting system has some desired behavior. The complication is that the behavior of the new system is described by a differential equation and we need the solution to that differential equation to understand how the system behaves. During the design process we will want to know the behavior of the resulting closed-loop system every time we make changes to the controller. This means we may need to solve for the solution of the differential equation many times. Being able to solve for the solution of the differential equation, x(t), or at least approximate x(t) is central to control systems design. Fortunately, control system theory provide short-cuts for approximating the solution to the differential equation.

In a linear system, the solution to the differential equation can be divided into two part: transient solution and steady-state solution. The transient solution describes the behavior of the system near time = 0. It is how the system responds to changes in the input or how the system responds to non-zero initial conditions. The steady-state solution describes the behavior of the system for time >> 0. Control system approximations are based on the solutions to each of these parts of the differential equation.

Transient Solution using DEQ

Given a differential equation of the form

x(n)+an-1x(n-1)+an-2x(n-2)+ ... +a1x(1) + +a0x     
     = bn-1y(n-1)+bn-2y(n-2)+ ... +b1y(1) + +b0y
2.2.1

where x(n) = dnx/dtn

The transient solution can be found by setting the input y(t)=0and assuming a solution has the formand assuming a form for the solution.

xtransient(t)=Aeλt
2.2.2

This leads to the "characteristic equation." (See a text on differential equations for details. img)

anλn + an-1λn-1 + ... + a1λ + a0 = 0
2.2.3
The characteristic equation is an nth order polynomial. The equation has n roots . These will be either real or complex conjugate pairs and can be written as (c+di) where d=0 for real roots. From DEQ theory we know that the total transient solution to the differential equation is the linear combination of all possible solutions given by Eqn 2.2.3. Therefore the transient solution has the form
xtransient(t)= nj=1 Aj e(c+di)t
2.2.4

where n is the order of the system, c+di is the ith root the character equation of the system, and Ai is some constant. When there are repeated roots the form is slightly different so refer DEQ text for details.

Using Eulers identity

This complex form is not very useful in engineering but Euler identity tells us that

eθ i=cos(θ)+i sin(θ)
2.2.5

Substitute Eqn 2.2.5 into Eqn 2.2.4 and following through with the algebra leads to the form (3)

or
xtransient(t)= nj=1 Aj e(cj)t sin(dj t+ϕ)
2.2.6
where ϕ is a phase angle resulting from combining terms from complex conjugate pairs.

This equation tells us that the xtransient is the sum of terms where each term of the solution has three factors,

  1. Aj is a constant and is unknown until we complete the solutions. Fortunately we can say a lot about the solution without knowing Ai.
  2. ect is a decaying exponential if c< 0, a value of 1 if c = 0, and a growing exponential if c > 0.
  3. sin(dt+ϕ) is a sinusoidal with frequency of d. If the roots of the characteristic equation are real then d and ϕ are zero and this factor has a value of 1 (no oscillation).

Remember, the goal is to approximate, or characterize, the solution to a differential equation. The first step is understanding how each term of the solution acts. Figure 2.2.1 illustrates how these factors combine to create one term of the transient solution. The graph show the response for a non-zero initial condition. Experiment with different values of c (the value in the exponent) and d (the frequency).


xtransient(t) = А e(ci+d)t = A e-0.1 t sin(
1.0
t+ϕ)
Figure 2.2.1. Transient response of a DEQ

Questions


  1. Which parameter determines how long it takes for the response to die out (answer)
    The real component, c, determines how long it takes for the oscillations to die out. This is time is related to the "settling time" discussed later.
  2. What determines the frequency of the system (answer)
    The frequency is determined by d, the imaginary component.
  3. What is the response when the real, c, and imaginary, d, components have the same absolute value? (Try c=-0.5, d= 0.5 and c=-2.0, d=2.0) (answer)
    The system response has one "bump" with a small amount of undershoot. Later we will learn that this corresponds to a "damping ratio" of 0.707.
  4. What happens to the response when the real part, c, is positive? (answer)
    The system response grows larger. If a system has any roots where the real component is greater than zero, the system is called "unstable". In control system the a system must have all roots with negative real components or "all roots must be in the left half plane" of the complex plane.

The total transient solution is the sum of many of these terms. The key is that the control systems designer does not need to know the exact time domain response, but instead needs to characterize the response. In Section 2.3 we will use parameters such as frequency of oscillation, amount of overshoot, damping and settling time to help us characterize the response.

Transient Solution using the Transfer Function

We can perform the same analysis using a transfer function. Given the transfer function

X(s)____________Y(s)
=
bn-1 sn-1 + bn-2 sn-2 + ... + b1s + b0____________an sn + an-1 sn-1+ ... + a1s + a0
2.2.7
In controls systems the terms "poles of the system", "roots of the denominator of the transfer function" and "roots of the characteristic equation" are interchangeable. Both characterize the transient response of the differential equation.

The term "zeros" refers to the roots of the numerator of the transfer function (the b terms).

Notice that the denominator of the transfer function is the characteristic equation of the differential equation except it is written in s instead of λ. Thus the roots of the denominator, also called the poles of the system (tell me more), are the roots of the characteristic equation for the differential equation. Its common in control systems design to refer to the "poles" of the system, instead of the roots of the characteristic equation.

roots of    an sn + an-1 sn-1+ ... + a1s + a0=0 → poles of the system
2.2.8

Approximating the Transient Response

The poles of the system, or roots of the characteristic equation of the differential system, characterize the transient response of the system. In control systems design the poles are typically plotted as complex numbers on the complex plane. With practice you should be able to look at a plot of the characteristic equation and approximate the transient solution to the differential equation. Practice approximating the solution for differential differential equation using the following experiment.

img Transient Response of a DEQ

Steady-State Solution

The steady-state solution describes the behavior of the system for time >> 0. The steady-state solution can be found by assuming the steady-state response of the system follows the same form as the input. This means if the input, y(t), is sinusoidal, the output will be sinusoidal. (remind me)

When a differential equation is linear and has constant coefficients, as we have limited our systems to, the steady-state, or particular solution, can be found using the Method of Undetermined Coefficients. In the M.U.C. you assume the solution has the same form as the input, guess at the solution and solve for the unknown constants. Go here img for details

To characterize the steady-state solution, you must specify the input to the system. In control systems we typically limit these inputs to step input, ramp inputs and sinusoidal inputs. In this section we will limit our discussion to step inputs.

One key is that the differential equations considered on this site are all linear and so superposition applies. That is, the response to multiple inputs is the sum of the responses to each of the individual inputs; and the response to a scaled input is the response to the unscaled output multiplied by the scaling factor.


Step Input

A step input, μ(t), is defined as a function such that

μ(t) = 0 for t < 0
μ(t) = 1 for t ≥ 0
2.2.9
We are interested in the system response when the input y(t)= μ(t). From differential equations we know that x(t) will have the same form as y(t) for large t. That is, if y(t) is constant, x(t) will be some constant once it reaches steady-state. It then follows that for large t that dx/dt=0, d2x/dt2=0… and dy/dt=0, d2y/dt2=0… (since the derivative of a constant is zero).

Substituting μ(t) for y(t)in Eqn 2.1, letting t → ∞ and setting the derivatives to zero we have

lim x(t) t→∞ = b0μ(∞)/a0 = b0/a0
2.2.10

Step Input Transfer Function

We can perform the same analysis using a transfer function. Given the transfer function

X(s)____________Y(s)
=
bn-1 sn-1 + bn-2 sn-2 + ... + b1s + b0____________an sn + an-1 sn-1+ ... + a1s + a0
2.2.11
Let y(t)= μ(t) where
ℒ μ(t) = 1/s
2.2.12
and apply the Final Value Theoremimg
lim g(t) t→∞ = lim s→0 s(G(s))
2.2.13

Substituting the step function, Eqn 2.2.12, in for Y(s) in Eqn 2.2.11

X(s) =
bn-1 sn-1 + bn-2 sn-2 + ... + b1s + b0____________an sn + an-1 sn-1+ ... + a1s + a0
×
1____________s
2.2.14
apply the final value theorem
For a step input:
lim x(t) t→∞ = lim s→0 s(X(s)) = b0/a0
2.2.15
Poles are the roots of the characteristic equation or the roots of the denominator of the transfer functions.

Zeros are the roots of the numerator of the transfer function.

Combining the Transient and Steady-State Response

The total response of a differential equation is a combination of the transient and steady-state solution. Again, the goal is to be able to approximate, or characterize, the system response with out solving for the time domain response of the differential equation (or transfer function). In controls systems design the poles and zeros (remind me) of the system are plotted on the complex plane. By simply analyzing the pole and zero location the designer is able to characterize the response of the system. In the following experiment you can adjust the poles and zeros of a transfer function and see how they affect the transient and steady-state step response.

img Total Response of a DEQ