\min_{u} \sum_{i=1}^{N} \|y_{k+i|k}-r_{k+i}\|_{Q_i}^2 + \|Δu_{k+i|k}\|_{R_i}^2

The expression you've provided appears to be an optimization problem where the goal is to minimize the cost function involving the system states and control inputs. Let's break down the components:

\[
\min_{u} \sum_{i=1}^{N} \|y_{k+i|k} - r_{k+i}\|_{Q_i}^2 + \|Δu_{k+i|k}\|_{R_i}^2
\]

### Components:

1. **Objective:** The goal is to minimize the sum of two terms over the index \(i\) from \(1\) to \(N\).
   - **First term:** \( \|y_{k+i|k} - r_{k+i}\|_{Q_i}^2 \)
     - Here, \(y_{k+i|k}\) represents the predicted state of the system at time \(k + i\), given the information at time \(k\).
     - \(r_{k+i}\) is a reference or desired trajectory for the state at time \(k + i\).
     - The notation \( \| \cdot \|_{Q_i}^2 \) indicates a weighted squared norm, where \(Q_i\) is a weighting matrix or scalar that penalizes deviations between the predicted state and the reference.
     
   - **Second term:** \( \|Δu_{k+i|k}\|_{R_i}^2 \)
     - \( \Delta u_{k+i|k} \) represents the change in the control input between time steps \(k + i - 1\) and \(k + i\).
     - The norm \( \| \cdot \|_{R_i}^2 \) indicates the cost of the control effort (or its rate of change), weighted by the matrix or scalar \(R_i\).

### Objective:  
You are minimizing a combination of:
- The deviation of the system states from the desired reference trajectory over time, penalized by \(Q_i\).
- The change in the control input over time, penalized by \(R_i\).

This is typically seen in optimal control problems, like **Model Predictive Control (MPC)**, where \(y\) is the predicted state, \(r\) is the reference, and \(Δu\) represents control input changes.

Would you like more details on how to solve this problem or any specific steps in optimization or control theory related to this? Or do you have any additional parts of the problem you'd like to discuss?

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### Related Questions:
1. What are some common approaches to solve this type of optimization problem in MPC?
2. How do the matrices \(Q_i\) and \(R_i\) affect the system's performance and control behavior?
3. What is the role of the prediction horizon \(N\) in this optimization?
4. How would you formulate the problem if the system dynamics were non-linear?
5. How can this objective be expanded to handle constraints on the state or control inputs?

### Tip: 
In MPC, a common approach is to use a **recursion** to solve this optimization problem at each time step by predicting the system behavior and adjusting the control inputs accordingly.

min_{u} \sum_{i=1}^{N} \|y_{k+i|k}-r_{k+i}\|_{Q_i}^2 + \|Δu_{k+i|k}\|_{R_i}^2

Learn about the optimization problem involved in Model Predictive Control (MPC), which minimizes a cost function that combines state deviations and control input changes, penalized by weight matrices Q_i and R_i. This problem involves key concepts in control theory, optimization, and quadratic cost functions, suitable for graduate-level studies in control engineering.

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