When choosing state variables for a system, they must be linearly independent. What does it mean for a set of variables to be linearly independent?
Explanation
Linear independence is a crucial requirement for state variables. It ensures that each variable provides unique information about the system's state, and no variable is redundant. If a variable could be expressed as a combination of others, it would not be needed in the minimal set.
Other questions
What is the primary disadvantage of the classical, or frequency-domain, approach to system analysis and design?
What is the term for the smallest set of linearly independent system variables whose values at time t0, along with known forcing functions, completely determine the value of all system variables for all time greater than or equal to t0?
A state-space representation consists of two key equations. What are they?
In the general state-space representation x_ = Ax + Bu, what does the matrix 'A' represent?
What is the typical minimum number of state variables required to describe a system represented by a differential equation?
What is a convenient method for selecting state variables when modeling electrical networks?
For a transfer function C(s)/R(s) = 24 / (s^3 + 9s^2 + 26s + 24), what is the third row of the system matrix A when represented in phase-variable form?
What is the formula used to convert a state-space representation (A, B, C, D) back into a transfer function T(s)?
In the translational mechanical system of Example 3.3, with masses M1 and M2, spring K, and damper D, what is the state equation for dv2/dt?
What is the key mathematical tool used to linearize nonlinear state equations about an equilibrium point?
For the simple pendulum in Example 3.7, what is the nonlinear state equation for x_2, the angular acceleration?
For the Antenna Control case study, the transfer function of the power amplifier is G(s) = 100/(s + 100). What is its state-space representation if the state variable is the output, ea?
In the Pharmaceutical Drug Absorption case study, what is the state equation for dx3/dt, the rate of change of the drug amount in the blood?
In the vector-matrix state equation x_ = Ax + Bu, the input u is a vector. What type of system does this general form allow for that the classical single transfer function does not easily handle?
For the electrical network in Skill-Assessment Exercise 3.1, what is the value of the A[1,3] element in the system matrix?
How is a transfer function with a numerator polynomial, such as (b2*s^2 + b1*s + b0), handled when converting to a state-space representation?
For the system in Example 3.6, with state matrix A = [[0, 1, 0], [0, 0, 1], [-1, -2, -3]], what is the determinant of (sI - A)?
In the RLC network shown in Figure 3.2, the state variable representation in Eq. (3.15) uses x_ = Ax + Bu. What is the element A[2,2] of the system matrix A?
What is the n-dimensional space whose axes are the state variables called?
In Skill-Assessment Exercise 3.2, for a translational mechanical system with three masses, how many state variables are used in the state-space representation?
When can a state variable, such as the voltage across a resistor vR(t), NOT be chosen as part of the minimal set of state variables if the current i(t) is already chosen?
Given the transfer function G(s) = (2s + 1) / (s^2 + 7s + 9) from Skill-Assessment Exercise 3.3, what is the correct phase-variable state-space representation?
In the Antenna Control case study, the final state equation for the motor and load has a system matrix A = [[0, 1], [0, -1.71]]. What physical quantities do the two state variables, x1 and x2, represent?
What is the result of the state equation x_ = Ax + Bu if the input u is zero?
In the Case Study on pharmaceutical drug absorption, the system of five state equations has an input matrix B that is a null vector. How can the system have a response?
For the system represented by the transfer function G(s) = (s^2 + 7s + 2) / (s^3 + 9s^2 + 26s + 24), what is the output equation vector C in phase-variable form?
In the electrical network of Example 3.2 (with a dependent source), the relationship vL = vC + iR2*R2 is used. How is iR2 expressed in terms of other circuit variables at Node 2?
Why is the state-space representation of a system not unique?
What is the primary role of the output equation, y = Cx + Du, in a state-space representation?
For a translational mechanical system as in Example 3.3, if M1=2, M2=1, D=3, and K=4, what is the value of the A[2,2] element in the system matrix?
In the general state-space representation y = Cx + Du, what does a non-zero D matrix imply?
If a system is described by the differential equation d^3y/dt^3 + 5*d^2y/dt^2 + 8*dy/dt + 2y = 10u, what is the B vector in the phase-variable state-space representation?
In the RLC circuit of Figure 3.2, with state variables q and i, the output is chosen as the inductor voltage, vL(t). What is the correct output equation?
For the system in Skill-Assessment Exercise 3.4, with state matrix A = [[-4, -1.5], [4, 0]], what is the characteristic polynomial, det(sI - A)?
An 8th-order system would be represented in state space with how many state equations?
In the state-space representation of the pharmaceutical drug absorption model, what is the element A[3,4] of the system matrix?
If you have a transfer function where the numerator is a constant (e.g., 24) and the denominator is s^3 + 9s^2 + 26s + 24, what is the output vector C in the phase-variable representation?
What is meant by the 'phase-variable' choice for state variables?
In the electrical network of Example 3.1, the state variable vC is the voltage across the capacitor. What is the state equation for dvC/dt?
For the system in Skill-Assessment Exercise 3.5, representing a mass with a nonlinear spring, what is the value of the A[2,1] element of the linearized system matrix?
If a system has three independent energy-storage elements, what is the minimum number of state variables required?
In the vector-matrix output equation y = Cx + Du, what is the state vector x?
For the mechanical system in Example 3.3, if M1=1, M2=1, D=2, and K=3, what is the value of A[2,3] in the system matrix?
When representing a translational mechanical system in state space, as in Example 3.3, what is a common choice for the state variables?
What is the physical interpretation of the state-vector trajectory in state space?
In the case study of the antenna control system's motor and load, the equivalent inertia Jm is 0.03 and the value of Kt/Ra is 0.0625. What is the value of the B[2,1] element in the state-space representation?
For the linearized pendulum in Example 3.7, if T represents a small perturbation torque, what is the linearized state equation for the change in angular acceleration, d(delta_x2)/dt?
If a system is represented in phase-variable form with A=[[0,1,0],[0,0,1],[-10,-5,-2]], what is the original differential equation for the output y=x1?
In the pharmaceutical drug model, the output is defined as the vector y containing all five state variables, x1 through x5. What would be the correct C matrix for the output equation y = Cx?