System Dynamics (24.509)

III. The State Space Equations and Their Time Domain Solution

Analytic Solution of Linear Stationary Systems

Now our next logical step is to address the time domain solution of the state equations. Our approach to developing analytic solutions will be to emphasize the analogy between the scalar problem and the matrix or the multivariable problem. We will see that the matrix exponential function plays an important role in developing solutions for linear stationary systems.

Homogeneous Linear Stationary Systems

Consider an LTI system which has no independent forcing function and only a single state variable. This results in a scalar homogeneous system.

Scalar Case

Assume a solution of the form

Substitution into the original equation gives

Now evaluating at t = 0 gives that . Therefore, the final solution is

Matrix Case

By analogy to the scalar case, assume a solution of the form

Substitution into the original equation gives

Now, using the initial conditions gives . Therefore, the final solution is

This result can also be cast into several other useful forms. For example, if we evaluate the initial condition at some arbitrary instead of t = 0, eqn. (3.29) becomes

Or writing , we have

and, substituting t for , we see that

is also a valid form of the solution. This form is particularly useful since is simply a constant matrix for constant. Letting , eqn. (3.31) can be written as

Thus the evaluation of requires only repeated matrix-vector multiplication. This discrete recursive formulation can also be written as .

In the technical literature, the matrix exponential, , is sometimes referred to as the state transition matrix. To see how this term applies, consider the following development.

In general, the solution to the homogeneous state equation can be written as

where is the state transition matrix which is the unique solution of

To show that the solution to eqn. (3.34) is indeed a solution to the original system of equations, consider the following:

and

For a homogeneous linear time-invariant system we know that , and we have just argued that . Therefore, the state transition matrix is simply . Thus, we see from eqn. (3.33) that the state transition matrix simply specifies a transformation of the initial conditions.

We already know several properties of the state transition matrix, . For example,

1.

2.

3.

4. Letting , gives

5. if

These properties can be written in terms of . For example, properties #3 and #4 can be written as

and

The terminology and manipulation of the so-called state transition matrix, , is important, but it is nothing more than working with the familiar matrix exponential.

Non-Homogeneous Linear Stationary Systems

We now address the solution of non-homogeneous systems, where there is an independent input function that drives the system response.

Scalar Case

Multiplying by the integrating factor, , gives

Integrating this expression between t_{0} and t gives

Finally, multiplying through by e^{at} and rearranging gives

If , this becomes

The standard procedure for checking the correctness of a solution [such as eqn. (3.37)] is to substitute it into the original differential equation. This case is no different, but one must be careful when differentiating integral terms that have variable limits of integration. The usual technique for doing this is called Leibnitz's Rule, which can be stated as follows:

If , then with suitable conditions on a(t), b(t), and [i.e. that they are well behaved with no discontinuities], we have

Substituting eqn. (3.37) into eqn. (3.35) and using Leibnitz's Rule, we have

Matrix Case

By analogy to the scalar case, multiply this expression by the integrating factor, , giving

Integrating this between t_{0} and t, one has

Finally, multiplying through by and rearranging gives

If , this becomes

As a check on this solution we can differentiate eqn. (3.41), giving

As a final note, one should be aware that these expressions can also be written in terms of the state transition matrix, where

For example, eqn. (3.41) written with becomes

A Note About Time-Varying Systems

In general, a closed form solution in terms of the matrix exponential is not possible for time varying systems. This is because, for the general case, do not commute. We will bypass this proof and refer the interested student to the literature (see your textbook by Ogata, for example) for further discussion of analytic solutions to time varying systems.

We have seen that linear stationary systems have analytical closed form solutions and that linear non-stationary systems (in general) do not. However, solutions to realistic problems with several unknowns or state variables must be determined via computer implementation, whether we are dealing with stationary or non-stationary systems. The analytic solutions for linear stationary systems are extremely important, but they can be generated by hand for only very low-order systems. Thus, no matter what form the system takes, computer implementation for realistic engineering problems will always be required.

24.509 Lecture Notes by Dr. J. R. White, UMass-Lowell (Spring 1997).