[10.2 INTRODUCTION TO](#page-14-0) STATE SPACE

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10.2 INTRODUCTION TO STATE SPACE

From the discussion in Ch. 1, we know that to determine a system’s response(s) at any instant t, we need to know the system’s inputs during its entire past, from −∞ to t. If the inputs are known only for t > t0, we can still determine the system output(s) for any t > t0, provided we know certain initial conditions in the system at t = t0. These initial conditions collectively are called the initial state of the system (at t = t0).

The state variables q1(t),q2(t),…,qN(t) are the minimum number of system variables such that their initial values at any instant t⁰ are sufficient to determine the behavior of the system for all time t ≥ t⁰ when the input(s) to the system is known for t ≥ t0. This statement implies that an output of a system at any instant is determined completely from a knowledge of the values of the system state and the input at that instant.

Initial conditions of a system can be specified in many different ways. Consequently, the system state can also be specified in many different ways. This means that state variables are not unique.

914 CHAPTER 10 STATE-SPACE ANALYSIS

This discussion is also valid for multiple-input, multiple-output (MIMO) systems, where every possible system output at any instant t is determined completely from a knowledge of the system state and the input(s) at the instant t. These ideas should become clear from the following example of an RLC circuit.

EXAMPLE 10.2 State-Space Description and Output Equations of an RLC Circuit

Find a state-space description of the RLC circuit shown in Fig. 10.1. Verify that all possible system outputs at some instant t can be determined from knowledge of the system state and the input at that instant t.

Figure 10.1 Circuit for Ex. 10.2.

It is known that inductor currents and capacitor voltages in an RLC circuit can be used as one possible choice of state variables. For this reason, we shall choose q¹ (the capacitor voltage) and q² (the inductor current) as our state variables.

The node equation at the intermediate node is

$$i_3 = i_1 - i_2 - q_2$$

but i³ = 0.2q˙1, i¹ = 2(x −q1), i² = 3q1. Hence,

$$0.2\dot{q}_1 = 2(x - q_1) - 3q_1 - q_2$$

or

$$\dot{q}_1 = -25q_1 - 5q_2 + 10x$$

This is the first state equation. To obtain the second state equation, we sum the voltages in the extreme right loop formed by C, L, and the 2 resistor so that they are equal to zero:

$$-q_1 + \dot{q}_2 + 2q_2 = 0$$ $$\quad \text{or} \quad$$

Thus, the two state equations are

$$\dot{q}_1 = -25q_1 - 5q_2 + 10x$$ $$\dot{q}_2 = q_1 - 2q_2$$

q˙² = q¹ −2q²

Every possible output can now be expressed as a linear combination of q1, q2, and x. From Fig. 10.1, we have

$$v_1 = x - q_1$$

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$$i_1 = 2(x - q_1)$$

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$$v_2 = q_1$$

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$$i_2 = 3q_1$$

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$$i_3 = i_1 - i_2 - q_2 = 2(x - q_1) - 3q_1 - q_2 = -5q_1 - q_2 + 2x$$

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$$i_4 = q_2$$

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$$v_4 = 2i_4 = 2q_2$$

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$$v_3 = q_1 - v_4 = q_1 - 2q_2$$

This set of equations is known as the output equation of the system. It is clear from this set that every possible output at some instant t can be determined from knowledge of q1(t), q2(t), and x(t), the system state, and the input at the instant t. Once we have solved the state equations to obtain q1(t) and q2(t), we can determine every possible output for any given input x(t).

For continuous-time systems, the state equations are N simultaneous first-order differential equations in N state variables q1, q2, … , qN of the form

$$\dot{q}_i = g_i(q_1, q_2, \dots, q_N, x_1, x_2, \dots, x_j)$$

$i = 1, 2, \dots, N$

where x1, x2, … , xj are the j system inputs. For a linear system, these equations reduce to a simpler linear form

$$\dot{q}_i = a_{i1}q_1 + a_{i2}q_2 + \dots + a_{iN}q_N + b_{i1}x_1 + b_{i2}x_2 + \dots + b_{ij}x_j \qquad i = 1, 2, \dots, N \qquad (10.14)$$

If there are k outputs y1, y2,…, yk, the k output equations are of the form

$$y_m = c_{m1}q_1 + c_{m2}q_2 + \dots + c_{mN}q_N + d_{m1}x_1 + d_{m2}x_2 + \dots + d_{mj}x_j \qquad m = 1, 2, \dots, k \quad (10.15)$$

The N simultaneous first-order state equations are also known as the normal-form equations.

These equations can be written more conveniently in matrix form:

$$\begin{bmatrix}\n\dot{q}_1 \\ \dot{q}_2 \\ \vdots \\ \dot{q}_N\n\end{bmatrix} = \begin{bmatrix}\na_{11} & a_{12} & \cdots & a_{1N} \\ a_{21} & a_{22} & \cdots & a_{2N} \\ \vdots & \vdots & \cdots & \vdots \\ a_{N1} & a_{N2} & \cdots & a_{NN}\n\end{bmatrix} \begin{bmatrix}\nq_1 \\ q_2 \\ \vdots \\ q_N\n\end{bmatrix} + \begin{bmatrix}\nb_{11} & b_{12} & \cdots & b_{1j} \\ b_{21} & b_{22} & \cdots & b_{2j} \\ \vdots & \vdots & \ddots & \vdots \\ b_{N1} & b_{N2} & \cdots & b_{Nj}\n\end{bmatrix} \begin{bmatrix}\nx_1 \\ x_2 \\ \vdots \\ x_j\n\end{bmatrix}$$

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$$\begin{bmatrix}\ny_1 \\ y_2 \\ \vdots \\ y_k\n\end{bmatrix} = \begin{bmatrix}\nc_{11} & c_{12} & \cdots & c_{1N} \\ c_{21} & c_{22} & \cdots & c_{2N} \\ \vdots & \vdots & \ddots & \vdots \\ c_{k1} & c_{k2} & \cdots & c_{kN}\n\end{bmatrix} \begin{bmatrix}\nq_1 \\ q_2 \\ \vdots \\ q_N\n\end{bmatrix} + \begin{bmatrix}\nd_{11} & d_{12} & \cdots & d_{1j} \\ d_{21} & d_{22} & \cdots & d_{2j} \\ \vdots & \vdots & \cdots & \vdots \\ d_{k1} & d_{k2} & \cdots & d_{kj}\n\end{bmatrix} \begin{bmatrix}\nx_1 \\ x_2 \\ \vdots \\ x_j\n\end{bmatrix}$$

or

and

q˙ = Aq+Bx (10.16)

and

$$y = Cq + Dx \tag{10.17}$$

Equation (10.16) is the state equation and Eq. (10.17) is the output equation; q, y, and x are the state vector, the output vector, and the input vector, respectively.

For discrete-time systems, the state equations are N simultaneous first-order difference equations. Discrete-time systems are discussed in Sec. 10.7.