1.10 INTERNAL [DESCRIPTION: THE](#page-8-0) STATE-SPACE DESCRIPTION

← Back to LINEAR SYSTEMS AND SIGNALS Overview

1.10 INTERNAL DESCRIPTION: THE STATE-SPACE DESCRIPTION

We shall now introduce the state-space description of a linear system, which is an internal description of a system. In this approach, we identify certain key variables, called the state variables, of the system. These variables have the property that every possible signal in the system can be expressed as a linear combination of these state variables. For example, we can show that every possible signal in a passive RLC circuit can be expressed as a linear combination of independent capacitor voltages and inductor currents, which, therefore, are state variables for the circuit.

To illustrate this point, consider the network in Fig. 1.43. We identify two state variables: the capacitor voltage q¹ and the inductor current q2. If the values of q1, q2, and the input x(t) are known at some instant t, we can demonstrate that every possible signal (current or voltage) in the circuit can be determined at t. For example, if q¹ = 10, q² = 1, and the input x = 20 at some instant, the remaining voltages and currents at that instant will be

$$i_1 = (x - q_1)/1 = 20 - 10 = 10 \text{A}$$

\n

$$v_1 = x - q_1 = 20 - 10 = 10 \text{V}$$

\n

$$v_2 = q_1 = 10 \text{V}$$

\n

$$i_2 = q_1/2 = 5 \text{A}$$

\n

$$i_C = i_1 - i_2 - q_2 = 10 - 5 - 1 = 4 \text{A}$$

\n

$$i_3 = q_2 = 1 \text{A}$$

\n

$$v_3 = 5q_2 = 5 \text{V}$$

\n

$$v_L = q_1 - v_3 = 10 - 5 = 5 \text{V}$$

\n(1.33)

Thus all signals in this circuit are determined. Clearly, state variables consist of the key variables in a system; a knowledge of the state variables allows one to determine every possible output of the system. Note that the state-variable description is an internal description of a system because it is capable of describing all possible signals in the system.

Figure 1.43 Choosing suitable initial conditions in a network.

EXAMPLE 1.20 State-Space Description of a System

This example illustrates how state equations may be natural and easier to determine than other descriptions, such as loop or node equations. Consider again the network in Fig. 1.43 with q¹ and q² as the state variables and write the state equations.

This can be done by simple inspection of Fig. 1.43. Since q˙¹ is the current through the capacitor,

$$\dot{q}_1 = i_C = i_1 - i_2 - q_2$$

= $(x - q_1) - 0.5 q_1 - q_2$
= $-1.5 q_1 - q_2 + x$

Also 2q˙2, the voltage across the inductor, is given by

$$2\dot{q}_2 = q_1 - v_3$$ $$= q_1 - 5q_2$$

or

$$\dot{q}_2 = 0.5 q_1 - 2.5 q_2$$

Thus, the state equations are

$$\dot{q}_1 = -1.5q_1 - q_2 + x \n\dot{q}_2 = 0.5q_1 - 2.5q_2$$

\n(1.34)

This is a set of two simultaneous first-order differential equations. This set of equations comprises the state equations. Once these equations have been solved for q¹ and q2, everything else in the circuit can be determined by using Eq. (1.33), which are known as the output equations. Thus, in this approach, we have two sets of equations, the state equations and the output equations. Once we have solved the state equations, all possible outputs can be obtained

from the output equations. In the input–output description, an Nth-order system is described by an Nth-order equation. In the state-variable approach, the same system is described by N simultaneous first-order state equations.†

EXAMPLE 1.21 Controllability and Observability

Investigate the nature of state equations and the issue of controllability and observability for the circuit in Fig. 1.41a.

This circuit has only one capacitor and no inductors. Hence, there is only one state variable, the capacitor voltage q(t). Since C = 1 F, the capacitor current is q˙. There are two sources in this circuit: the input x(t) and the capacitor voltage q(t). The response due to x(t), assuming q(t) = 0, is the zero-state response, which can be found from Fig. 1.44a, where we have shorted the capacitor [q(t) = 0]. The response due to q(t) assuming x(t) = 0, is the zero-input response, which can be found from Fig. 1.44b, where we have shorted x(t) to ensure x(t) = 0. It is now trivial to find both the components.

Figure 1.44a shows zero-state currents in every branch. It is clear that the input x(t) sees an effective resistance of 5 , and, hence, the current through x(t) is x/5 A, which divides in the two parallel branches, resulting in the current x/10 through each branch.

Examining the circuit in Fig. 1.44b for the zero-input response, we note that the capacitor voltage is q and the current is q˙. We also observe that the capacitor sees two loops in parallel, each with resistance 4 and current q˙/2. Interestingly, the 3 branch is effectively shorted because the circuit is balanced, and thus the voltage across the terminals cd is zero. The total current in any branch is the sum of the currents in that branch in Figs. 1.44a and 1.44b (principle of superposition).

Branch

\n

$$c = \frac{x}{10} + \frac{\dot{q}}{2} \quad 2\left(\frac{x}{10} + \frac{\dot{q}}{2}\right)$$

\n

$$cb = \frac{x}{10} - \frac{\dot{q}}{2} \quad 2\left(\frac{x}{10} - \frac{\dot{q}}{2}\right)$$

\n

$$ad = \frac{x}{10} - \frac{\dot{q}}{2} \quad 2\left(\frac{x}{10} - \frac{\dot{q}}{2}\right)$$

\n

$$bd = \frac{x}{10} + \frac{\dot{q}}{2} \quad 2\left(\frac{x}{10} + \frac{\dot{q}}{2}\right)$$

\n

$$ec = \frac{x}{5} \quad 3\left(\frac{x}{5}\right)$$

\n

$$ed = \frac{x}{5} \quad x$$

\n(1.35)

^† This assumes the system to be controllable and observable. If it is not, the input–output description equation will be of an order lower than the corresponding number of state equations.

Figure 1.44 Analysis of a system that is neither controllable nor observable.

To find the state equation, we note that the current in branch ca is (x/10)+ ˙q/2 and the current in branch cb is (x/10)− ˙q/2. Hence, the equation around the loop acba is

$$q = 2\left[-\frac{x}{10} - \frac{\dot{q}}{2}\right] + 2\left[\frac{x}{10} - \frac{\dot{q}}{2}\right] = -2\dot{q}$$ $$\dot{q} = -0.5q$$

(1.36)

or

This is the desired state equation.

Substitution of q˙ = −0.5q in Eq. (1.35) shows that every possible current and voltage in the circuit can be expressed in terms of the state variable q and the input x, as desired. Hence, the set of Eq. (1.35) is the output equation for this circuit. Once we have solved the state equation [Eq. (1.36)] for q, we can determine every possible output in the circuit.

The output y(t) is given by

$$y(t) = 2\left[\frac{x}{10} - \frac{\dot{q}}{2}\right] + 2\left[\frac{x}{10} + \frac{\dot{q}}{2}\right] = \frac{2}{5}x(t)$$

(1.37)

A little examination of the state and the output equations indicates the nature of this system. Equation (1.36) shows that the state q(t) is independent of the input x(t); hence the system state q cannot be controlled by the input. Moreover, Eq. (1.37) shows that the output y(t) does not depend on the state q(t). Thus, the system state cannot be observed from the output terminals. Hence, the system is neither controllable nor observable. Such is not the case of other systems examined earlier. Consider, for example, the circuit in Fig. 1.43. The state equations [Eq. (1.34)] show that the states are influenced by the input directly or indirectly. Hence, the system is controllable. Moreover, as Eq. (1.33) shows, every possible output is expressed in terms of the state variables and the input. Hence, the states are also observable.

State-space techniques are useful not just because of their ability to provide internal system description, but for several other reasons, including the following.

• 1. State equations of a system provide a mathematical model of great generality that can describe not just linear systems, but also nonlinear systems; not just time-invariant systems, but also time-varying parameter systems; not just SISO (single-input/single-output) systems, but also multiple-input/multiple-output (MIMO) systems. Indeed, state equations are ideally suited for the analysis, synthesis, and optimization of MIMO systems.
• 2. Compact matrix notation and the powerful techniques of linear algebra greatly facilitate complex manipulations. Without such features, many important results of the modern system theory would have been difficult to obtain. State equations can yield a great deal of information about a system even when they are not solved explicitly.
• 3. State equations lend themselves readily to digital computer simulation of complex systems of high order, with or without nonlinearities, and with multiple inputs and outputs.
• 4. For second-order systems (N = 2), a graphical method called phase-plane analysis can be used on state equations, whether they are linear or nonlinear.

The real benefits of the state-space approach, however, are realized for highly complex systems of large order. Much of the book is devoted to introduction of the basic concepts of linear systems analysis, which must necessarily begin with simpler systems without using the state-space approach. Chapter 10 deals with the state-space analysis of linear, time-invariant, continuous-time, and discrete-time systems.

DR ILL 1.20 State Equations for a Series RLC Circuit

Write the state equations for the series RLC circuit shown in Fig. 1.45, using the inductor current q1(t) and the capacitor voltage q2(t) as state variables. Express every voltage and current in this circuit as a linear combination of q1, q2, and x.

ANSWERS

q¹ = −3q¹ −q² +x and q² = 2q1.

Figure 1.45 Circuit for Drill 1.20.