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Modern Control Systems

Dorf

Chapter 11

The Design of State Variable Feedback Systems - all with Video Answers

Educators

Chapter Questions

Problem 1

A first-order system is represented by the timedomain differential equation
$$
\dot{x}=x+u .
$$

A feedback controller is to be designed such that
$$
u(t)=-k x,
$$
and the desired equilibrium condition is $x(t)=0$ as $t \rightarrow \infty$. The performance integral is defined as
$$
J=\int_0^{\infty} x^2 d t,
$$
and the initial value of the state variable is $x(0)=\sqrt{2}$. Obtain the value of $k$ in order to make $J$ a minimum. Is this $k$ physically realizable? Select a practical value for the gain $k$ and evaluate the performance index with that gain. Is the system stable without the feedback due to $u(t)$ ?

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Problem 2

To account for the expenditure of energy and resources, the control signal is often included in the performance integral. Then the operation will not involve an unlimited control signal $u(t)$. One suitable performance index, which includes the effect of the magnitude of the control signal, is
$$
J=\int_0^{\infty}\left(x^2(t)+\lambda u^2(t)\right) d t .
$$
(a) Repeat Problem P11.1 for the performance index.
(b) If $\lambda=2$, obtain the value of $k$ that minimizes the performance index. Calculate the resulting minimum value of $J$.

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Problem 3

An unstable robot system is described by the vector differential equation [10]
$$
\frac{d}{d t}\left[\begin{array}{l}
x_1 \\
x_2
\end{array}\right]=\left[\begin{array}{rr}
1 & 0 \\
-1 & 2
\end{array}\right]\left[\begin{array}{l}
x_1 \\
x_2
\end{array}\right]+\left[\begin{array}{l}
1 \\
1
\end{array}\right] u(t) .
$$

Both state variables are measurable, and so the control signal is set as $u(t)=-k\left(x_1+x_2\right)$. Following the method of Section 11.7, design gain $k$ so that the performance index is minimized. Evaluate the minimum value of the performance index. Determine the sensitivity of the performance to a change in $k$. Assume that the initial conditions are
$$
\mathbf{x}(0)=\left[\begin{array}{l}
1 \\
1
\end{array}\right] .
$$
Is the system stable without the feedback signals due to $u(t)$ ?

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Problem 4

Determine the feedback gain $k$ of Example 11.12 that minimizes the performance index
$$
J=\int_0^{\infty} \mathbf{x}^T \mathbf{x} d t
$$
when $\mathbf{x}^T(0)=\left[\begin{array}{ll}1 & 1\end{array}\right]$. Plot the performance index $J$ versus the gain $k$.

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Problem 5

Determine the feedback gain $k$ of Example 11.13 that minimizes the performance index
$$
J=\int_0^{\infty}\left(\mathbf{x}^T \mathbf{x}+\mathbf{u}^T \mathbf{u}\right) d t
$$
when $\mathbf{x}^T(0)=\left[\begin{array}{ll}1 & 1\end{array}\right]$. Plot the performance index $J$ versus the gain $k$.

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Problem 6

For the solutions of Problems P11.3, P11.4, and P11.5, determine the roots of the closed-loop optimal control system. Note that the resulting closed-loop roots depend on the performance index selected.

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03:33

Problem 7

A system has the vector differential equation as given in Equation (11.42). We want both state variables to be used in the feedback so that $u(t)=-k_1 x_1-k_2 x_2$. Also, we desire to have a natural frequency $\omega_n$ for this system equal to 2 . Find a set of gains $k_1$ and $k_2$ in order to achieve an optimal system when $J$ is given by Equation (11.63). Assume $\mathbf{x}^T(0)=\left[\begin{array}{ll}1 & 0\end{array}\right]$.

James Kiss
James Kiss
Numerade Educator
00:37

Problem 8

For the system of Example 11.11 determine the optimum value for $k_2$ when $k_1=1$ and $\mathbf{x}^T(0)=\left[\begin{array}{ll}1 & 0\end{array}\right]$.

Victor Salazar
Victor Salazar
Numerade Educator
06:03

Problem 9

An interesting mechanical system with a challenging control problem is the ball and beam, shown in Figure P11.9(a) [11]. It consists of a rigid beam that is free to rotate in the plane of the paper around a center pivot, with a solid ball rolling along a groove in the top of the beam. The control problem is to position the ball at a desired point on the beam using a torque applied to the beam as a control input at the pivot.
A linear model of the system with a measured value of the angle $\phi$ and its angular velocity $d \phi / d t=\omega$ is available. Select a feedback scheme so that the response of the closed-loop system has an overshoot of $4 \%$ and a settling time (with a $2 \%$ criterion) of 1 second for a step input.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:07

Problem 10

The dynamics of a rocket are represented by
$$
\begin{aligned}
& \dot{\mathbf{x}}=\left[\begin{array}{ll}
0 & 0 \\
1 & 0
\end{array}\right] \mathbf{x}+\left[\begin{array}{l}
1 \\
0
\end{array}\right] u \\
& y=\left[\begin{array}{ll}
0 & 1
\end{array}\right] \mathbf{x}
\end{aligned}
$$
and state variable feedback is used, where $u=-8 x_1$ $16 x_2$. Determine the roots of the characteristic equation of this system and the response of the system when the initial conditions are $x_1(0)=1$ and $x_2(0)=0$.

James Kiss
James Kiss
Numerade Educator

Problem 11

The state variable model of a plant to be controlled is
$$
\begin{aligned}
& \dot{\mathbf{x}}=\left[\begin{array}{rr}
-5 & -2 \\
2 & 0
\end{array}\right] \mathbf{x}+\left[\begin{array}{c}
0.5 \\
0
\end{array}\right] u \\
& y=\left[\begin{array}{ll}
0 & 1
\end{array}\right] \mathbf{x}+[0] u .
\end{aligned}
$$

Use state variable feedback and incorporate a command input $u=-\mathbf{K x}+\alpha r$. Select the gains $\mathbf{K}$ and $\alpha$ so that the system has a rapid response with an overshoot of approximately $1 \%$, a settling time (with a $2 \%$ criterion) less than 1 second, and a zero steady-state error to a unit step input.

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Problem 12

A DC motor has the state variable model
$$
\begin{aligned}
& \dot{\mathbf{x}}=\left[\begin{array}{rrlrr}
-3 & -2 & -0.75 & 0 & 0 \\
-3 & 0 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 2 & 0
\end{array}\right] \mathbf{x}+\left[\begin{array}{l}
1 \\
0 \\
0 \\
0 \\
0
\end{array}\right] u \\
& y=\left[\begin{array}{lllll}
0 & 0 & 0 & 0 & 2.75
\end{array}\right] \mathbf{x} .
\end{aligned}
$$

Determine whether this system is controllable and observable.

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Problem 13

Consider a system with state-space model
$$
\begin{aligned}
& \dot{\mathbf{x}}=\left[\begin{array}{rrr}
0 & 1 & 0 \\
0 & 0 & 1 \\
-2 & -3 & -5
\end{array}\right] \mathbf{x}+\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right] u \\
& y=x_1
\end{aligned}
$$

Design a state feedback controller such that closed-loop poles are $s_1=-4.8, s_2=-4.8+j 6, s_3=-4.8-j 6$.

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Problem 14

A process has the transfer function
$$
\begin{aligned}
& \dot{\mathbf{x}}=\left[\begin{array}{cc}
-10 & 0 \\
1 & 0
\end{array}\right] \mathbf{x}+\left[\begin{array}{l}
1 \\
0
\end{array}\right] u \\
& y=\left[\begin{array}{ll}
0 & 1
\end{array}\right] \mathbf{x}+[0] u .
\end{aligned}
$$

Determine the state variable feedback gains to achieve a settling time (with a $2 \%$ criterion) of 1 second and an overshoot of about $10 \%$. Also sketch the block diagram of the resulting system. Assume the complete state vector is available for feedback.

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Problem 15

A telerobot system has the matrix equations [18]
$$
\dot{\mathbf{x}}=\left[\begin{array}{rrr}
-1 & 0 & 0 \\
0 & -2 & 0 \\
0 & 0 & -3
\end{array}\right] \mathbf{x}+\left[\begin{array}{l}
1 \\
1 \\
0
\end{array}\right] u
$$
and
$$
y=\left[\begin{array}{lll}
1 & 0 & 2
\end{array}\right] \mathbf{x} .
$$
(a) Determine the transfer function, $G(s)=$ $Y(s) / U(s)$. (b) Draw the block diagram indicating the state variables. (c) Determine whether the system is controllable. (d) Determine whether the system is observable.

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Problem 16

Hydraulic power actuators were used to drive the dinosaurs of the movie Jurassic Park [23]. The motions of the large monsters required high-power actuators requiring 1200 watts.
One specific limb motion has dynamics represented by
$$
\begin{aligned}
& \dot{\mathbf{x}}=\left[\begin{array}{rr}
-2 & 0 \\
1 & 0
\end{array}\right] \mathbf{x}+\left[\begin{array}{l}
1 \\
0
\end{array}\right] u \\
& y=\left[\begin{array}{ll}
0 & 1
\end{array}\right] \mathbf{x}+[0] u .
\end{aligned}
$$

We want to place the closed-loop poles at $s=$ $-2 \pm j 2$. Determine the required state variable feedback using Ackermann's formula. Assume that the complete state vector is available for feedback.

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03:39

Problem 17

A system has a transfer function
$$
\frac{Y(s)}{R(s)}=\frac{s+a}{s^4+9 s^3+28 s^2+38 s+24} .
$$

Determine a real value of $a$ so that the system is either uncontrollable or unobservable.

Arpit Gupta
Arpit Gupta
Numerade Educator

Problem 18

A system has a plant
$$
\frac{Y(s)}{U(s)}=G(s)=\frac{1}{(s+1)^2} .
$$
(a) Find the matrix differential equation to represent this system. Identify the state variables on a block diagram model. (b) Select a state variable feedback structure using $u(t)$, and select the feedback gains so that the response $y(t)$ of the unforced system is critically damped when the initial condition is $x_1(0)=1$ and $x_2(0)=0$, where $x_1=y(t)$. The repeated roots are at $s=-\sqrt{2}$.

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Problem 19

The block diagram of a system is shown in Figure P11.19. Determine whether the system is controllable and observable.

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04:15

Problem 20

Consider the automatic ship-steering system discussed in Problems P8.11 and P9.15. The state variable form of the system differential equation is
$$
\dot{\mathbf{x}}(t)=\left[\begin{array}{cccr}
-0.05 & -6 & 0 & 0 \\
-10^{-3} & -0.15 & 0 & 0 \\
1 & 0 & 0 & 13 \\
0 & 1 & 0 & 0
\end{array}\right] \mathbf{x}(t)+\left[\begin{array}{c}
-0.2 \\
0.03 \\
0 \\
0
\end{array}\right] \delta(t),
$$
where $\mathbf{x}^T(t)=\left[\begin{array}{llll}v & \omega_s & y & \theta\end{array}\right]$. The state variables are $x_1=v=$ the transverse velocity; $x_2=\omega_s=$ angular rate of ship's coordinate frame relative to response frame; $x_3=y=$ deviation distance on an axis perpendicular to the track; $x_4=\theta=$ deviation angle. (a) Determine whether the system is stable. (b) Feedback can be added so that
$$
\delta(t)=-k_1 x_1-k_3 x_3 .
$$

Determine whether this system is stable for suitable values of $k_1$ and $k_3$.

Manish Jain
Manish Jain
Numerade Educator
04:20

Problem 21

An $R L$ circuit is shown in Figure P11.21. (a) Select the two stable variables and obtain the vector differential equation where the output is $v_0(t)$. (b) Determine whether the state variables are observable when $R_1 / L_1=R_2 / L_2$. (c) Find the conditions when the system has two equal roots.
Figure can't copy

Ashley Boni
Ashley Boni
Numerade Educator

Problem 22

A manipulator control system has a loop transfer function of
$$
G(s)=\frac{1}{s(s+0.4)}
$$
and negative unity feedback [17]. Represent this system by a state variable signal-flow graph or block diagram and a vector differential equation. (a) Plot the response of the closed-loop system to a step input. (b) Use state variable feedback so that the overshoot is $5 \%$ and the settling time (with a $2 \%$ criterion) is 1.35 seconds. (c) Plot the response of the state variable feedback system to a step input.

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04:10

Problem 23

Consider again the system of Example 11.7 when we desire that the steady-state error for a step input be zero and the desired roots of the characteristic equation be $s=-2 \pm j 1$ and $s=-10$.

James Kiss
James Kiss
Numerade Educator

Problem 24

Consider again the system of Example 11.7 when we desire that the steady-state error for a ramp input be zero and the roots of the characteristic equation be $s=-2 \pm j 2$ and $s=-20$.

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Problem 25

Consider the system represented in state variable form
$$
\begin{aligned}
& \dot{\mathbf{x}}=\mathbf{A x}+\mathbf{B} u \\
& y=\mathbf{C x}+\mathbf{D} u,
\end{aligned}
$$
where
$$
\begin{aligned}
& \mathbf{A}=\left[\begin{array}{rr}
1 & 4 \\
-5 & 10
\end{array}\right], \quad \mathbf{B}=\left[\begin{array}{l}
0 \\
1
\end{array}\right], \\
& \mathbf{C}=\left[\begin{array}{ll}
1 & -4
\end{array}\right], \quad \text { and } \quad \mathbf{D}=[0] .
\end{aligned}
$$

Verify that the system is observable. Then design a full-state observer by placing the observer poles at $s_{1,2}=-1$. Plot the response of the estimation error $\mathbf{e}=\mathbf{x}-\hat{\mathbf{x}}$ with an initial estimation error of $\mathbf{e}(0)=\left[\begin{array}{ll}1 & 1\end{array}\right]^T$.

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01:33

Problem 26

Consider the third-order system
$$
\begin{aligned}
& \dot{\mathbf{x}}=\left[\begin{array}{rrr}
0 & 1 & 0 \\
0 & 0 & 1 \\
-1 & -2 & -3
\end{array}\right] \mathbf{x}+\left[\begin{array}{l}
0 \\
0 \\
4
\end{array}\right] u \\
& y=\left[\begin{array}{lll}
2 & -4 & 0
\end{array}\right] \mathbf{x}+[0] u .
\end{aligned}
$$

Verify that the system is observable. If so, determine the observer gain matrix required to place the observer poles at $s_{1,2}=-1 \pm j 2$ and $s_3=-10$.

James Kiss
James Kiss
Numerade Educator
04:52

Problem 27

Consider the second-order system
$$
\begin{aligned}
& \dot{\mathbf{x}}=\left[\begin{array}{rr}
1 & 0 \\
-3 & -2
\end{array}\right] \mathbf{x}+\left[\begin{array}{c}
10 \\
0
\end{array}\right] u \\
& y=\left[\begin{array}{ll}
1 & 0
\end{array}\right] \mathbf{x}+[0] u .
\end{aligned}
$$

Determine the observer gain matrix required to place the observer poles at $s_{1,2}=-1 \pm j$.

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator

Problem 28

Consider the single-input, single-output system is described by
$$
\begin{aligned}
\dot{\mathbf{x}}(t) & =\mathbf{A x}(t)+\mathbf{B} u(t) \\
y(t) & =\mathbf{C} \mathbf{x}(t)
\end{aligned}
$$
where
$$
\mathbf{A}=\left[\begin{array}{cc}
0 & 1 \\
-16 & -8
\end{array}\right], \mathbf{B}=\left[\begin{array}{l}
0 \\
K
\end{array}\right], \mathbf{C}=\left[\begin{array}{ll}
1 & 0
\end{array}\right] .
$$
(a) Determine the value of $K$ resulting in a zero steady-state tracking error when $u(t)$ is a unit step input for $t \geq 0$. The tracking error is defined here as $e(t)=u(t)-y(t)$.
(b) Plot the response to a unit step input and verify that the tracking error is zero for the gain $K$ determined in part (a).

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Problem 29

The block diagram shown in Figure P11.29 is an example of an interacting system. Determine a state variable representation of the system in the form
$$
\begin{aligned}
\dot{\mathbf{x}}(t) & =\mathbf{A x}(t)+\mathbf{B} u(t) \\
y(t) & =\mathbf{C x}(t)+\mathbf{D} u(t)
\end{aligned}
$$
Figure can't copy

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