$$

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$$H(s) = \frac{4s}{(s+1)(s+2)^2}$$

\n(b)

\n

$$H(s) = \frac{s^3 + 7s^2 + 12s}{(s+1)^3(s+2)}$$

10.3-1 Find the state vector q(t) by using the Laplace transform method if

$$\dot{\mathbf{q}} = \mathbf{A}\mathbf{q} + \mathbf{B}\mathbf{x}$$

where

$$\mathbf{A} = \begin{bmatrix} 0 & 2 \\ -1 & -3 \end{bmatrix} \qquad \mathbf{B} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$$ $$\mathbf{q}(0) = \begin{bmatrix} 2 \\ 1 \end{bmatrix} \qquad x(t) = 0$$

10.3-2 Repeat Prob. 10.3-1 for

$$\mathbf{A} = \begin{bmatrix} -5 & -6 \\ 1 & 0 \end{bmatrix} \qquad \mathbf{B} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$$ $$\mathbf{q}(0) = \begin{bmatrix} 5 \\ 4 \end{bmatrix} \qquad \qquad x(t) = \sin 100t$$

10.3-3 Repeat Prob. 10.3-1 for

$$\mathbf{A} = \begin{bmatrix} -2 & 0 \\ 1 & -1 \end{bmatrix} \qquad \mathbf{B} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$$ $$\mathbf{q}(0) = \begin{bmatrix} 0 \\ -1 \end{bmatrix} \qquad x(t) = u(t)$$

10.3-4 Repeat Prob. 10.3-1 for

$$\mathbf{A} = \begin{bmatrix} -1 & 1 \\ 0 & -2 \end{bmatrix} \qquad \mathbf{B} = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$$ $$\mathbf{q}(0) = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \qquad \mathbf{x} = \begin{bmatrix} u(t) \\ \delta(t) \end{bmatrix}$$

10.3-5 Use the Laplace transform method to find the response y for

$$\dot{\mathbf{q}} = \mathbf{A}\mathbf{q} + \mathbf{B}\mathbf{x}(t)$$ $$y = \mathbf{C}\mathbf{q} + \mathbf{D}\mathbf{x}(t)$$

where

$$\mathbf{A} = \begin{bmatrix} -3 & 1 \\ -2 & 0 \end{bmatrix} \qquad \mathbf{B} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$$ $$\mathbf{C} = \begin{bmatrix} 0 & 1 \end{bmatrix} \qquad \mathbf{D} = 0$$

and

$$x(t) = u(t) \qquad \mathbf{q}(0) = \begin{bmatrix} 2 \\ 0 \end{bmatrix}$$

10.3-6 Repeat Prob. 10.3-5 for

$$\mathbf{A} = \begin{bmatrix} -1 & 1 \\ -1 & -1 \end{bmatrix} \quad \mathbf{B} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$$ $$\mathbf{C} = \begin{bmatrix} 1 & 1 \end{bmatrix} \quad \mathbf{D} = 1$$ $$x(t) = u(t) \quad \mathbf{q}(0) = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$$

!

10.3-7 The transfer function H(s) in Prob. 10.2-8 is realized as a cascade of H1(s) followed by H2(s), where

$$H_1(s) = \frac{1}{s+3}$$ $$H_2(s) = \frac{3s+10}{s+4}$$

Let the outputs of these subsystems be state variables q¹ and q2, respectively. Write the state equations and the output equation for this system and verify that H(s) = Cφ(s)B+D.

• 10.3-8 Find the transfer function matrix H(s) for the system in Prob. 10.3-5.
• 10.3-9 Find the transfer function matrix H(s) for the system in Prob. 10.3-6.
• 10.3-10 Find the transfer function matrix H(s) for the system

$$\dot{q} = Aq + Bx$$ $$y = Cq + Dx$$

where

$$\mathbf{A} = \begin{bmatrix} 0 & 1 \\ -1 & -2 \end{bmatrix} \quad \mathbf{B} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \quad \mathbf{x} = \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix}$$ $$\mathbf{C} = \begin{bmatrix} 1 & 2 \\ 4 & 1 \\ 1 & 1 \end{bmatrix} \qquad \mathbf{D} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 1 & 0 \end{bmatrix}$$

• 10.3-11 Repeat Prob. 10.3-1, using the time-domain method.
• 10.3-12 Repeat Prob. 10.3-2, using the time-domain method.
• 10.3-13 Repeat Prob. 10.3-3, using the time-domain method.
• 10.3-14 Repeat Prob. 10.3-4, using the time-domain method.
• 10.3-15 Repeat Prob. 10.3-5, using the time-domain method.
• 10.3-16 Repeat Prob. 10.3-6, using the time-domain method.
• 10.3-17 Find the unit impulse response matrix h(t) for the system in Prob. 10.3-7, using Eq. (10.45).
• 10.3-18 Find the unit impulse response matrix h(t) for the system in Prob. 10.3-6.
• 10.3-19 Find the unit impulse response matrix h(t) for the system in Prob. 10.3-10.
• 10.4-1 The state equations of a certain system are given as

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

Define a new state vector w such that

$$w_1 = q_2$$ $$w_2 = q_2 - q_1$$

Find the state equations of the system with w as the state vector. Determine the characteristic roots (eigenvalues) of the matrix A in the original and the transformed state equations.

10.4-2 The state equations of a certain system are

$$\dot{q}_1 = q_2 \n\dot{q}_2 = -2q_1 - 3q_2 + 2x$$

(a) Determine a new state vector w (in terms of vector q) such that the resulting state equations are in diagonalized form.

(b) For output y given by

$$y = Cq + Dx$$

where

$$\mathbf{C} = \begin{bmatrix} 1 & 1 \\ -1 & 2 \end{bmatrix} \qquad \mathbf{D} = 0$$

determine the output y in terms of the new state vector w.

10.4-3 Given a system

$$\dot{\mathbf{q}} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & -2 & -3 \end{bmatrix} \mathbf{q} + \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} x$$

determine a new state vector w such that the state equations are diagonalized.

10.4-4 The state equations of a certain system are given in diagonalized form as

$$\dot{\mathbf{q}} = \begin{bmatrix} -1 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & -2 \end{bmatrix} \mathbf{q} + \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} x$$

The output equation is given by

$$y = \begin{bmatrix} 1 & 3 & 1 \end{bmatrix} \mathbf{q}$$

Determine the output y for

$$\mathbf{q}(0) = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} \qquad x(t) = u(t)$$

10.5-1 Write the state equations for the systems depicted in Fig. P10.5-1. Determine a new state vector w such that the resulting state equations are in diagonalized form. Write the output y in terms of w. Determine in each case whether the system is controllable and observable.

Figure P10.5-1

10.6-1 An LTI discrete-time system is specified by

2

!

$$\mathbf{A} = \begin{bmatrix} 2 & 0 \\ 1 & 1 \end{bmatrix} \qquad \mathbf{B} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$$ $$\mathbf{C} = \begin{bmatrix} 0 & 1 \end{bmatrix} \qquad \mathbf{D} = \begin{bmatrix} 1 \end{bmatrix}$$

and

• q(0) = 1 x[n] = u[n]
• (a) Find the output y[n], using the timedomain method.
• (b) Find the output y[n], using the frequencydomain method.
• 10.6-2 An LTI discrete-time system is specified by the difference equation

$$y[n+2] + y[n+1] + 0.16y[n]$$

= $x[n+1] + 0.32x[n]$

• (a) Show the DFII, its transpose, cascade, and parallel realizations of this system.
• (b) Write the state and the output equations from these realizations, using the output of each delay element as a state variable.
• 10.6-3 Repeat Prob. 10.6-2 for

$$y[n+2] + y[n+1] - 6y[n]$$

= 2x[n+2] + x[n+1]

• 10.7-1 Verify the state and output equations for the LTID system shown in Fig. 10.15.
• 10.7-2 Verify the state and output equations for the LTID system shown in Fig. 10.16.