Signals and Systems Lecture 16

More Inverse Laplace

Transforms & Transfer Functions

May 5, 2008

Today’s Topics

1. Inverse Laplace transforms for higher order systems 2. LTI system Transfer functions and block diagrams 3. Feedback control systems

Take Away

The operator calculus enabled by Laplace transforms can be very useful for analyzing feedback control systems.

Required Reading

O&W-9.4, 9.8

Last time we saw that a partial fraction expansion for determining inverse Laplace transforms can be performed using graphical methods. We are interested in the entire range of rational Laplace transforms and we know that they are made up of both first and second order terms, especially second order terms with damping ratios less than one. As an example consider the following transfer function that includes a first order pole at the origin and a second order term with a damping ratio less than one.

This equation can also be written as

and the three system poles are

!

s1,

!

s2 and

!

s3

The s plane pole/zero configuration is

In order to take the inverse Laplace transform we will perform a partial fraction expansion

Each of the constants, which we also call residues, is determined by multiplying H(s) by the pole associated with that constant and evaluating the result at the pole location. This is the cover up method. For example the

!

K1

term is

Alternatively, we can use the graphical approach from last time, as depicted in the following diagram

So the denominator terms are

and

!

K1 is evaluated as

which is the same answer as earlier. Applying the graphical method to find

!

K2 yields the following diagram

where now

and

!

K2 is

Finally, since

!

K3 must be the complex conjugate of

!

K2

Applying these results to the original partial fraction expansion yields

Thus we have a sum of first order terms in s, so we can readily obtain the inverse Laplace transform.

or

Transfer Functions As we saw earlier, rational Laplace transforms are typically obtained by transforming a LTI differential equation. For example, suppose we have a differential equation of the form

where u(t) is the input and y(t) is the output. Then take the Laplace transform of both sides of this equation, assuming all initial conditions are zero, to obtain

The relationship between the input and the output (i.e., the transfer function) is then readily determined by taking the quotient of their Laplace transforms.

For any LTI system the transfer function completely characterizes the input/output dynamical characteristics of the system. Typically, we draw an input/output block diagram to represent it as follows

The system’s response to any input can be readily obtained by taking the Laplace transform of the input and multiplying by the system transfer function. The result is the Laplace transform of the system output. Then the time domain representation of the output can be readily obtained by taking the inverse Laplace transform of Y(s). The block diagram representation that we have been discussing provides a very powerful algebraic approach for characterizing and analyzing more complex systems. Often there can be elaborate interconnections of many subsystem elements that together constitute a total system. Aircraft and spacecraft avionics and guidance and control systems are good examples of such complex interconnected systems. For example, suppose there is a series connection of two systems. In the time domain it would be represented as

and in the Laplace domain as

The impulse response of this system can be obtained by convolution of the individual impulse responses

whereas its transfer function is merely a product

Alternatively, two systems may be connected in parallel

and in the Laplace domain

So the system transfer function is

Typically, the transfer function representations in the Laplace domain are so convenient to use and manipulate that most analysis and design of systems is performed using this methodology. In fact, at least in the early stages of a system design, the analyses to determine and compare the performances of tentative designs are almost exclusively performed in the Laplace domain rather than in the time domain. One very important configuration that is commonly represented using transfer functions is the classic feedback control configuration. It is typically represented in the time domain as

The primay objective of feedback control is to induce a system, which is typically called the plant, to follow, in some sense, input commands. The system input, or command signal, is x(t) and the output is y(t). The error e(t) is the difference between the input x(t) and a feedback signal z(t). The error e(t) is the input to the controller, which operates on the error to provide actuation to the plant. In most instances the controller acts as a power amplifier that turns the low power error signal into a much higher power input to the plant. The plant is the system that is to be controlled or commanded. The plant might, for example, be an aircraft or spacecraft and the control function might be to control the attitude or orientation of the vehicle. Commonly the feedback path consists of some kind of sensing mechanism that measures the output y(t) and produces the feedback signal z(t). The feedback signal is provided to the summer, to be subtracted from the input, to produce the error signal. The elements in the forward and feedback paths can be combined into transfer functions that represent the aggregation of all elements in the forward and feedback paths respectively. Thus the system diagram becomes

where

!

G(s) is the feedback path transfer function and

!

H(s) is the forward path transfer function, and

Also, the Laplace transforms of the input, output and error signal are related as follows

These three simultaneous linear equations can be combined so as to eliminate the error and feedback signal transforms, to realize the system input/output equation

and we immediately define the closed loop system transfer function as

We can immediately use this equation to illustrate one of the most important advantages of a feedback system. Suppose the feedback transfer function is unity, implying that the feedback sensing mechanism simply feeds back the output without any modification. Then the feedback system transfer function becomes

Now suppose that the magnitude of the forward path transfer function is very large, so

This might be accomplished by designing the controller element in the forward path to have a very high amplification of the error signal (i.e., high gain amplification). This causes the controller to act strongly to correct for any errors. Then, as a result

and

In effect the controller will drive the system to eliminate errors, thereby making the output follow the input with high fidelity. Furhermore, large forward path amplification, or gain, tends to make the closed loop system independent of the particular dynamics of the various other system elements, including the plant. This is the basic idea of feedback control. However, as is always the case in real world situations, there are significant limits and tradeoffs that must be made to implement this concept. The most significant of these limitations is usually system stability. If the forward path amplification becomes too high then typically the system can be driven into a regime of unstable operation, which is usually unsatisfactory or even dangerous. In the remaining few lectures we will explore these issues to a modest extent and in 16.06 the issue of system stability will be a major topic of study.