In Example 12.6, for a system with matrices A = [[0, 1, 0], [0, 0, 1], [-4, -3, -2]] and C = [0, 5, 1], what is the determinant of the observability matrix, OM?
Explanation
This question tests the ability to construct the observability matrix and evaluate its determinant to determine observability, as shown in Example 12.6. A non-zero determinant indicates the system's states can be inferred from its output.
Other questions
What is a primary advantage of state-space design methods over frequency domain methods, such as root locus?
In the context of controller design for a plant in phase-variable form, what is the fundamental principle behind pole placement?
For the plant G(s) = 20(s + 5) / [s(s + 1)(s + 4)], a controller is designed to yield a 9.5 percent overshoot and a 0.74 second settling time, resulting in a desired characteristic equation of s^3 + 15.9s^2 + 136.08s + 413.1 = 0. What is the required value for the feedback gain k3?
According to Section 12.3, what is the definition of a controllable system?
What is the correct formulation of the controllability matrix (CM) for an nth-order plant?
In Example 12.2, a system is given with state equation matrices A = [[-1, 1, 0], [0, -1, 0], [0, 0, -2]] and B = [[0], [1], [1]]. What is the determinant of its controllability matrix, CM?
For a system not in phase-variable form, a transformation z = Px is used. How is the transformation matrix P calculated to convert the system to phase-variable form for controller design?
In Example 12.4, a state-variable feedback controller is designed for a plant to yield a 20.8 percent overshoot and a settling time of 4 seconds. The third closed-loop pole is chosen to cancel the closed-loop zero at s = -4. What is the desired characteristic equation for the closed-loop system?
What is the primary purpose of an observer in a state-space control system design?
The dynamics of the estimation error, e_x, in a closed-loop observer are governed by which state equation?
In Example 12.5, an observer is designed for a plant. The observer is specified to respond 10 times faster than the controlled loop from Example 12.4, which had dominant poles at -1 +/- j2. The third observer pole is placed at -100. What is the desired characteristic polynomial for the observer?
What is the definition of an observable system?
What is the correct mathematical formulation of the observability matrix (OM) for an nth-order plant?
For designing an observer for a system not in observer canonical form, a transformation z = Px is used. How is the transformation matrix P calculated?
In Example 12.9, an observer is designed for a plant modeling blood glucose level. The desired transient response is described by a damping ratio of 0.7 and a natural frequency of 100. What is the value of the observer gain l1?
How is zero steady-state error for a step input typically achieved in state-space design?
In Example 12.10, an integral controller is designed for a plant with state equations defined by A = [[0, 1], [-3, -5]] and C = [1, 0]. The desired third-order characteristic polynomial is s^3 + 116s^2 + 1783.1s + 18,310 = 0. What is the required value for the integrator gain, Ke?
In the Antenna Control Case Study, a controller is designed to yield a 10 percent overshoot and a settling time of 1 second, with a third pole at -40. What is the desired characteristic equation for this controller design?
In the Antenna Control Case Study, based on the desired characteristic equation of s^3 + 48s^2 + 365.8s + 1832 = 0, what is the calculated value for the feedback gain k1?
For the observer design in the Antenna Control Case Study, the observer poles are specified to yield 10 percent overshoot and a natural frequency 10 times that of the controller's dominant pair (which was 6.77 rad/s). The observer's third pole is placed at -400. What is the characteristic equation for the observer?
Based on the observer design in the Antenna Control Case Study, which uses the desired characteristic equation from Eq. (12.140), what is the calculated value for the observer gain l1?
Which state-space form is noted in Section 12.2 as yielding the simplest evaluation of the feedback gains for controller design?
In a state-space system with state feedback, the control input u is defined by the equation u = r - Kx. What does the vector K represent?
When is pole placement considered a viable design technique for a controller?
What is the primary reason for designing an observer's response to be much faster than the controlled closed-loop system's response?
In a system with a diagonal system matrix and distinct eigenvalues, how can uncontrollability be identified by inspection?
In the controller design by matching coefficients in Example 12.3, for a plant Y(s)/U(s) = 10/[(s+1)(s+2)], the desired characteristic equation is s^2 + 16s + 239.5 = 0. What is the value of the feedback gain k2?
What is a significant disadvantage of state-space design methods mentioned in the introduction of Chapter 12?
After designing feedback gains Kx for a system transformed to phase-variable form, how is the gain vector Kz for the original system representation found?
What condition must be met for a system represented in parallel form with distinct eigenvalues to be observable by inspection?
In Example 12.7, a system has state matrices A = [[0, 1], [-5, -21/4]] and C = [5, 4]. What is the rank of its observability matrix, OM?
After designing an observer gain vector Lx for a system transformed to observer canonical form, how is the gain vector Lz for the original system representation found?
Which canonical form is most convenient for designing an observer?
When adding integral control to a state-space system, a new state variable, xN, is introduced. What does the derivative of this state variable, xN_dot, represent?
Consider the plant from Example 12.10, G(s) = 1/(s^2+5s+3). A controller without integral control is designed for a 10 percent overshoot. What is the steady-state error for a unit step input?
The design of state-variable feedback for closed-loop pole placement consists of equating the characteristic equation of the closed-loop system with a desired characteristic equation. For a plant in phase-variable form, what is the closed-loop system matrix?
In Skill-Assessment Exercise 12.1, for the plant G(s) = 100(s+10)/[s(s+3)(s+12)], the feedback gains are designed to yield 5 percent overshoot and a peak time of 0.3 second. What is the value of the feedback gain associated with the most significant coefficient of the characteristic equation?
What is the key difference between an open-loop observer and a closed-loop observer as described in Section 12.5?
In the Antenna Control Case Study, the plant transfer function is G(s) = 1325/[s(s^2 + 101.71s + 171)]. What is the system matrix A for this plant when represented in phase-variable form?
What must be the rank of the controllability matrix, CM, for an nth-order plant to be considered completely controllable?
In Skill-Assessment Exercise 12.2, a system is given by matrices A = [[-1, 1, 2], [0, -1, 5], [0, 3, -4]] and B = [[2], [1], [1]]. Is this system controllable?
What is the final step in determining observability if the observability matrix, OM, is a square matrix?
In the state-space representation of a system, what does the output equation y = Cx signify?
In Example 12.8, an observer is designed by transformation for a plant G(s) = 1/[(s+1)(s+2)(s+5)]. What is the determinant of the observability matrix for the original cascade form, OMz?
Why might a state-space design prove to be very sensitive to parameter changes?
According to the topology for pole placement described in Section 12.2, how many adjustable feedback gains are required for an nth-order system?
What is the steady-state error for the integral-controlled system designed in Example 12.10 for a unit step input?
In the Case Study observer design, the observer's characteristic equation is specified in Eq. 12.140. What is the value of the observer gain l3?
The final transfer function for the designed controller in Example 12.1 is T(s) = 20(s+5)/(s^3 + 15.9s^2 + 136.08s + 413.1). A simulation of this system shows a steady-state response that approaches 0.24 instead of unity. What does this indicate?