In deriving a mathematical model for a physical system one usually begins with a system of differential equations. It is often convenient to rewrite these equations as a system of first order differential equations. We will call this system of differential equations a state space representation. The solution to this system is a vector that depends on time and which contains enough information to completely determine the trajectory of the dynamical system. This vector is referred to as the state of the system, and the components of this vector are called the state variables. In order to illustrate these concepts consider the following two examples.

Example spring-mass-damper system

This is the same example discussed in the lecture #3 where we derived the mathematical model of the system. The equation describing the evolution of this system can be written as (see equation (2) in lecture #3):

Suppose that the input to the system is f and the output is x. In this problem we have a second order differential equation which means that we need two initial conditions to solve it uniquely. For example, specifying the value of x(to) and dx/dt(to) will allow for solution of x(t) for t>to. Therefore, we will also require two state variables to describe this system. We will label each state variable xi where i = 1,2 and make the assignments

Rewriting the differential equation above in terms of the state variables xi yields the following state-space description for the system:

We can use matrix notation to rewrite the equations in the above state space representation.

So for the above example, the state vector takes the form:

Example (2 mass)-spring-damper system

Click here to see the animation.

This is the same example discussed in the previous lecture where we derived the mathematical model of the system (set of differential equations). The equations describing the evolution of this system can be written as (see equations (2) and (3) from the previous lecture):

Suppose that the input to the system is f and the output is z. In this problem we have two second order differential equations which means that we need four initial conditions to solve them uniquely. For example, specifying the value of w(to), dw/dt(to), z(to), and dz/dt(to) will allow for solution of w(t) and z(t) for t>to. Therefore, we will also require four state variables to describe this system. We will label each state variable xi where i = 1, ...,4 and make the assignments

Rewriting the two differential equations above in terms of the state variables xi yields the following state-space description for the system:

We can use matrix notation to rewrite the equations in the above state space representation.

So for the above example, the state vector takes the form:

In general, the linear state space models assume the form:

where x(0) is given.
If the number of states is equal to n, number of inputs is equal to p and the number of outputs is equal to q, then A is an nxn square matrix, B is an nxp matrix, C is an qxn matrix, and D is an qxp matrix.

The previous examples illustrate how to go from a physical problem to a mathematical description in state space form. Pictorially:

We will usually want to choose the smallest set of state variables which will still fully describe the system. A system description in terms of the minimum number of state variables is called a minimal realization of the system. We will want next to compute the solution of the state space equation. This leads us to the study of Laplace transforms.