State Space Description – State and Output Equations

8.1 State Space Description – State and Output Equations

In this section we introduce the state space description of an N th order LTI system, which consists of a set of equations describing the relations among the input, output, and state:

State equation: x ′ (t ) = f (x(t), u(t), t) x[n +1] = f (x[n], u[n], n) (8.1.1b)

(8.1.1a)

Output equation: y(t) = g(x(t), u(t), t) y[n] = g(x[n], u[n], n) (8.1.2b)

State vector: x(t) = [x T (t ), · · · , x x[n] = [x 1 [n], · · · , x N [n]]

Input vector: u(t ) = [u T

1 (t ), · · · , u K (t )]

u[n] = [u 1 [n], · · · , u K [n]]

Output vector: y(t) = [y 1 (t ), · · · , y M (t )] T

y[n] = [y 1 [n], · · · , y M [n]] T

(cf.) Note that, in this chapter, the notation u(t )/u[n] denotes the general input

function, while the unit step function/sequence is denoted by u s (t )/u s [n].

Here, we have the definitions of the state and the state variable: Definition 8.1 State, State Variables, and State Vector

The state of a system at time t 0 is the amount of information at t 0 that, together with the input from t 0 , determines uniquely the behavior of the system for all t > t 0 . Note that the ‘behavior’ means all the responses, including the state, of the system. The state variables of a dynamic system are the variables forming the smallest set of variables which determine the state of the system. The state vector is composed of the state variables.

(Ex) For an RLC circuit driven by a source e(t ), the inductor current i L (t ) and capacitor voltage v C (t ) can form the state. The charge q C (t ) and inductor cur- rent i L (t ) can also make the state. It is because {i L (t ), v C (t )} or {q C (t ), i L (t )} can be determined for any t > t 0 if the value of input e(t ) is known for t > t 0 together with the initial condition {i L (t 0 ), v C (t 0 )} or {q C (t 0 ), i L (t 0 )}.

(Ex) For a moving body, the set of the position x(t) and velocity x ′ (t ) qualifies the state of the mechanical system since the information about x(t 0 ), x ′ (t 0 ), and u(t) (force) for t > t 0 is necessary and sufficient for determining x(t ) and x ′ (t ) for any time t > t 0 .

Especially, the state space descriptions of continuous-time/discrete-time LTI sys- tems are

8.1 State Space Description – State and Output Equations 363

State equation: x ′ (t ) = Ax(t) + Bu(t) x[n + 1] = Ax[n] + Bu[n] (8.1.3b)

(8.1.3a)

Output equation: y(t ) = Cx(t) + Du(t) y[n] = Cx[n] + Du[n] (8.1.4b)

(8.1.4a)

In Sect. 1.3.4, we illustrated how a continuous-time/discrete-time state diagram can be constructed for a given differential/difference equation. Once a state dia- gram is constructed, the corresponding state equation can easily be obtained by the following procedure:

1. Assign a state variable x i (t )/x i [n] to the output of each integrator s −1 /delay z −1 .

2. Write an equation for the input x ′ i (t )/x i [n + 1] of each integrator/delay.

3. Write an equation for each system output in terms of state variables and input(s).

Applying this procedure to the state diagrams in Figs. 1.19(a)/1.20(c) yields

y [n] = b 0 b 1 (8.1.6b) x 2 (t )

y (t ) = b 0 1 b 1 (8.1.6a)

x 2 [n]

which is referred to as the controllable canonical form. Also, for Figs. 1.19(b)/1.20(d), we obtain

) * ) *) * ) x * ′ 1 (t ) 0 −a 0 x 1 (t )

*) * ) = * + u (t ) (8.1.7a) x 1 [n + 1] 0 −a 0 x 1 [n] b 0 x ′ 2 (t )

[n] = 01 1 (8.1.8b) 2 (t )

(8.1.8a)

x 2 [n]

which is referred to as the observable canonical form. (cf.) Note that the controllable and observable canonical forms of state/output

equations are convenient for designing a controller and an observer, respec- tively. [F-1]

364 8 State Space Analysis of LTI Systems