Question

Can you design a system of the form \(\dot{x}=A x+B u, y=C x\), with a suitably chosen initial condition, such that the input \(u(t)=\sin t\) yields the output \(y(t)=\sin t\) ? (Note that if the output equation would have been of the form \(y=C x+D u\), the answer would be affirmative almost trivially; but the design requires \(D=0\).) If your answer is affirmative (which it should be) what conditions should the transfer function \(h(s)\) satisfy?

Short answer

Design matrices \( A, B, C \) to have poles at \( j, -j \); use \( x(0) = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \).

Step-by-step solution

Understanding the System

The system is described by the state-space equations: \( \dot{x} = Ax + Bu \) and \( y = Cx \). We need to design this system so that with \( u(t) = \sin t \), the output \( y(t) = \sin t \) as well.

Analyzing the Desired Output

We want the output \( y(t) = \sin t \), which means that \( \, y \) and \( \, u \) should have the same response characteristic. Therefore, the system should allow the input frequency to pass unchanged to the output.

Designing the Transfer Function

The transfer function \( h(s) \) of a linear time-invariant (LTI) system is given by \( h(s) = C(sI - A)^{-1}B \). Since \( y(t) = \sin t \) and the Laplace Transform of \( \sin t \) is \( \frac{1}{s^2 + 1} \), \( h(s) \) should have a pole at \( j \) and \( -j \), the imaginary unit.

Constructing the Transfer Function

To achieve the desired response, construct \( h(s) = \frac{1}{s^2 + 1} \). This implies that the characteristic polynomial of the system, which is \( \, sI - A \, \), should have eigenvalues \( j \) and \( -j \).

Selecting System Matrices

Based on the transfer function, select matrices: \( A = \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \), \( B = \begin{bmatrix} 0 \ 1 \end{bmatrix} \), and \( C = \begin{bmatrix} 1 & 0 \end{bmatrix} \). These choices ensure the system will generate output \( y(t) = \sin t \) from input \( u(t) = \sin t \).

Determine Initial Conditions

For the system to produce an output of \( \sin t \) with the specified dynamics, we can choose an initial condition \( x(0) = \begin{bmatrix} 0 \ 1 \end{bmatrix} \) so that it starts in a state which immediately responds as a \( \, \sin t \, \).

Key concepts

State-Space Representation

The state-space representation is vital in control theory as it provides a compact way to represent and analyze dynamic systems. Using matrices, this form expresses the system's state variables and inputs, offering a clear view of how the system evolves over time. In this format, the system dynamics are expressed through a set of first-order differential equations. For instance, in our exercise, the equations

• \( \dot{x} = Ax + Bu \)
• \( y = Cx \)

represent how the state \( x \) changes with time under the influence of input \( u \), while \( y \) denotes the output derived from the state.
This representation is particularly insightful since it captures all system dynamics in terms of matrix operations, making the solution of these systems systematic and straightforward.

Transfer Function

The transfer function is a powerful tool used in control theory to analyze linear time-invariant (LTI) systems in the frequency domain. It's derived from the Laplace transform of the state-space representation and shows the output response of the system to any given input signal. In our scenario, the transfer function is described by:

• \( h(s) = C(sI - A)^{-1}B \)

To satisfy the condition that input \( u(t) = \sin t \) produces an output \( y(t) = \sin t \), the transfer function needs specific poles corresponding to the signal's frequency. Here, poles at \( j \) and \( -j \) are required for the sinusoidal response. The proper construction of \( h(s) \) is crucial in ensuring that these characteristics are present so that the output reflects the expected sinusoidal behavior.

Linear Time-Invariant Systems

Linear time-invariant or LTI systems are systems whose behavior can be described by linear differential equations, and they remain constant over time. LTI systems are central in control theory due to their predictability and simplicity in analysis. Their key feature is that they obey the superposition principle, meaning the response to a combination of inputs is the sum of the responses to each input taken individually.

In the context of the design problem, the LTI nature of the system permits us to straightforwardly determine the system's behavior purely based on its mathematical properties without having to consider time-variant changes. By identifying our system as LTI, we can use techniques such as state-space representation and transfer functions to predict the system’s response to any input, ensuring, for example, that the system responds with \( y(t) = \sin t \) when given \( u(t) = \sin t \).

System Design

The design of dynamic systems targets creating a system that meets particular performance specifications. In this task, we crafted a system using matrices \( A \), \( B \), and \( C \) so that when the input is \( u(t) = \sin t \), the output is also \( y(t) = \sin t \). This process involved selecting suitable matrices that align with our desired transfer function.

• Choosing \( A = \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \) effectively sets eigenvalues at \( j \) and \( -j \), ensuring oscillatory behavior.
• Letting \( B = \begin{bmatrix} 0 \ 1 \end{bmatrix} \) allows the input to influence the system state directly.
• Setting \( C = \begin{bmatrix} 1 & 0 \end{bmatrix} \) extracts the correct combination of state variables to match our desired output.
• Finally, choosing an initial condition of \( x(0) = \begin{bmatrix} 0 \ 1 \end{bmatrix} \) ensures the system begins in a state aligned with our input behavior.

Thus, system design not only involves selecting proper matrices but also ensuring compatibility with initial states and desired dynamics, crafting a precise and effective control solution.