The tool DC-10 plots time response of a dynamic system in state space formulation. The system is subject to initial disturbance and five types
of inputs, i.e., impulse, step, ramp, exponential and sinusoidal inputs. The tool also gives the eigenvalues of the system.

In addition, a tutorial on how to use the tool in computation is provided and a subject review on fundamental theories is presented.

**Example: Time Response of State Space Formulation**

The input in the tool is

After clicking "Run", the results will be shown in another page.

And the eigenvalues of the system are also shown as follows.

The eigenvalues of the system are

-2.5 + i * (1.32287566)

-2.5 + i * (-1.32287566)

### State Representation

A linear time-invariant (LTI) multi-input-multi-output (MIMO) dynamic system can be represented in state space form

where x is the vector of state variables, u the vector of inputs, and y the vector of outputs. For a dynamic system with n state variables, p inputs and q outputs, the matrices and vectors in the previous equations are of the following dimensions:

The block diagram of the state representation is shown in Fig. 1.

**Figure 1.** The block diagram of state space formulation

**Example**: Consider a spring-mass-damper system in Figure 2

**Figure 2.** A spring-mass-damper system

The governing differential equation of the system is

Choose the state variables as

Let the external force f be the input. Assume that the outputs of the system are the displacement and the force by the damper, namely,

A state representation of the system then is

### Time Response

The time response of the previously described dynamic system is the solution of Eqs. (1) and (2) subject to initial condition

Analytically, the solution takes the form

where e^{At} is an exponential matrix that can be written as the infinite series

The solution of the state equation (1) can also be determined by approximate methods, such as finite difference methods and numerical integration algorithms. Rungeâ€“Kutta methods are commonly used iterative methods for solution of state equations.

### References

1. R.C. Dorf and R.H. Bishop, 2010, Modern Control Systems, 12th edition, Prentice Hall.

2. K. Ogata, 2009, Modern Control Engineering, 5th Edition, Prentice Hall.

3. C.T. Chen, 1999, Linear System Theory and Design, 3rd Edition, Oxford University Press.