Question

We are given the time discrete system $$ \begin{aligned} x(k+1) &=\left(\begin{array}{cc} 0 & 1 \\ -2 & -3 \end{array}\right) x(k)+\left(\begin{array}{l} 0 \\ 1 \end{array}\right) u(k) \\ y(k) &=\left(\begin{array}{cc} 2 & 1 \end{array}\right) x(k) \end{aligned} $$ Determine the transition matrix, impulse response function and the transfer function of this system. Suppose the following periodic input signal is applied to the system: $$ u(k)= \begin{cases}0 & k<0 \\ (-1)^{k}, & k \geq 0\end{cases} $$ What is the output response (take \(x(0)\) as the zero state)? Why is the output signal not periodic? The time-discrete, time-invariant, linear system characterized by matrices \((A, B, C, D)\) is called controllable if for each \(x_{0}, x_{1} \in \mathcal{R}^{n}\) a time \(k>0\) and a sequence \(u(0), u(1), \ldots\) exist such that \(x\left(k, x_{0}, u\right)=x_{1} .\) The meaning of \(x\left(k, x_{0}, u\right)\) will be clear; the state at time instant \(k\), starting with initial condition \(x(0)=x_{0}\) and having applied an input sequence \(u\). The system is observable if a \(k>0\) exists such that for any sequence of controls \(u\) we have: $$ y\left(j, x_{0}, u\right)=y\left(j, x_{1}, u\right), \quad j=0,1, \ldots, k, \text { implies } x_{0}=x_{1} $$ The conditions in terms of matrices \(A, B, C\) and \(D\) for controllability and observability are the same as in the time-continuous case. This will be shown in the next theorem for controllability. Sometimes one distinguishes null controllability \(\left(x_{1}=0\right)\) and reachability \(\left(x_{0}=0\right) .\) It can be shown that "standard" controllability, i.e. with arbitrary \(x_{0}\) and \(x_{1}\), is as strong as reachability (see also the proof of next theorem); the condition is: $$ \operatorname{rank}\left[B A B \ldots A^{n-1} B\right]=n $$ wheras null controllability is not as strong as "standard" controllability.

Short answer

The system output is not periodic despite periodic input due to the system dynamics breaking periodicity.

Step-by-step solution

Determine the State Transition Matrix

The state transition matrix, often denoted by \( oldsymbol{ heta}(k) \), is calculated using the matrix exponential \( oldsymbol{ heta}(k) = A^k \). For the given matrix \( A = \begin{pmatrix} 0 & 1 \ -2 & -3 \end{pmatrix} \), we write it as a companion matrix form and solve to find powers of \( A \). We also know \( oldsymbol{ heta}(0) = I \), the identity matrix.

Calculate Impulse Response Function

The impulse response function, \( h(k) \), of a discrete-time system is found using \( h(k) = C oldsymbol{ heta}(k-1) B \). Substitute \( C = \begin{pmatrix} 2 & 1 \end{pmatrix} \), \( oldsymbol{ heta}(k-1) \) from previous step, and \( B = \begin{pmatrix} 0 \ 1 \end{pmatrix} \) to find \( h(k) \).

Derive the Transfer Function

The transfer function is given by \( H(z) = C(zI - A)^{-1}B \). We substitute for \( C \), \( A \), and \( B \), and solve \( (zI - A) \) for its inverse then multiply accordingly. The final result represents the transfer function \( H(z) \) in terms of \( z \).

Determine System Response to Input

To determine the response of the system to a periodic input \( u(k) = (-1)^k \) for \( k \geq 0 \), use the zero-state response \( y \), which is calculated as a convolution of \( u(k) \) with \( h(k) \). Analyze how the convolution impacts periodicity.

Analyze the Periodicity of Output

The output is not necessarily periodic even if the input is periodic, due to the system dynamics characterized by matrix \( A \). The solution and step-wise character of system dynamics or state propagation due to the eigenvalues can break periodicity.

Key concepts

State Transition Matrix

The state transition matrix is a fundamental concept in the study of dynamical systems. For discrete time systems, it is often denoted by \( \Phi(k) \) and describes how the state of a system evolves over time, based on the system matrix \( A \). In this case, the state transition matrix is computed using the power of \( A \), such that \( \Phi(k) = A^k \).

To further simplify, let's consider the identity matrix \( I \), where \( \Phi(0) = I \). This starting point forms the backbone of the calculation for any time \( k \).

In practical terms:

• The matrix captures the dynamics without external inputs.
• Acts as a bridge between the initial state and the state at time \( k \).

Understanding the state transition matrix is crucial for predicting the behavior of complex systems and analyzing their stability.

Impulse Response Function

Impulse response functions are essential for understanding how a system reacts to sudden, momentary input - think of it as 'disturbance'. For discrete systems, it is defined as \( h(k) = C \Phi(k-1) B \). Essentially, it gives us a snapshot of the system's output response when \( u(k) \) is the unit impulse.

Key aspects to consider:

• \( C \) is the output matrix, transforming the state into an output.
• \( \Phi(k-1) \) reflects the state evolution from an impulse.
• \( B \) maps the input into the state space.

Understanding \( h(k) \) helps to model how inputs affect outputs and is integral when designing control systems. It allows engineers to forecast output behaviors and adjust systems for desired performance.

Transfer Function

Transfer functions provide a powerful tool for understanding the frequency domain characteristics of a system. The transfer function \( H(z) \) for a discrete system is derived from the Laplace transform. It is given by \( H(z) = C(zI - A)^{-1}B \).

Let's break it down:

• \( C \) transforms the state into an observable output.
• \( (zI - A)^{-1} \) characterizes the system dynamics in the frequency domain. Here \( zI \) subtracts \( A \) to capture system behavior.
• \( B \) illustrates how inputs affect the system state.

In essence, the transfer function translates a system's model into a format that highlights its responses across various frequencies, making it easier to design appropriate feedback for control systems.

Controllability

Controllability details the ability to move a system's state to a desired point in a finite amount of time using suitable inputs. In matrix terms, a system is controllable if the controllability matrix \( [B \, AB \, A^2B \, \ldots \, A^{n-1}B] \) has full rank, equal to the number of state variables \( n \).

Here are core elements of controllability:

• If the matrix has full rank, any state within the state space can be reached.
• Lack of full rank implies limitations in achieving certain states.
• Controllability precedes effective system control strategies.

It's an essential concept in ensuring system states can be manipulated using inputs, forming the foundation of modern control theory, including robot control and aerospace systems.

Observability

Observability reflects the ability to infer the internal states of a system solely from its outputs. Mathematically, a system is observable if its observability matrix \( [C^T \, (CA)^T \, (CA^2)^T \, \ldots \, (CA^{n-1})^T]^T \) has full rank. This concept reveals the system's state dynamics through output measurements alone.

Key points of observability:

• Full rank in the observability matrix means any state of the system can be determined from output data within finite time.
• It's crucial for designing systems where direct state measurement isn't feasible.
• Both controllability and observability are pillars of system design, ensuring effective control and stabilization.

This concept is crucial in sensor networks and automated systems where outputs are used to backtrack states for system monitoring and fault diagnosis.