Lecture 11 - Practical use of MPC

Resources:

---

---

MPC Algorithm

for t=0,1,2,...,infty do
	Determine x_t
	Solve open-loop opt. prob. with x_0=x
		-> u_0,u_1,u_2,...,u_(N-1)
	Set u_t=u_0
end

• To determine $x_t$ we can either measure it, estimate it or both (Kalman Filter).

Output Feedback MPC Controller

We copy $u_t$, feed it into the model of the system, get an estimate, and use the estimate $\hat{x}$ into the MPC controller.

A linear state estimator can be written as

$${\hat{x}}_{t+1}=A{\hat{x}}_t+B{u}_t+L(y_t-C{\hat{x}}_t)$$

Which is used in the Closed-loop Estimator - Luenberger Observer and Kalman Filter.

Reference Tracking (Regulation)

Typically we want the controlled variables to go to the reference

$${\gamma }_t\to {\gamma }_{ref}$$

Where ${\gamma }_t=H{x}_t$.

Note:

• Controlled variables ${\gamma }_t$ are often, but not always different from the measured variables $y_t$. Above we assumed that the measurement and estimate are equal.

When a system is Steady State, this means that (in the discrete case)

$$\begin{array}{r}{x}_s=A{x}_s+B{u}_s\\ \to x_s=(\mathbb{I}-A{)}^{-1}B{u}_s\end{array}$$

And

$${\gamma }_s=H{x}_s=H(\mathbb{I}-A{)}^{-1}B{u}_s$$

Offset-Free Control (=“Integral Action”)

Discussed in this paper.

In a PID controller, the effect of the integral is to remove unknown (constant) disturbances. However, there is no equivalent in the MPC, so how do we implement this?

---

We create a model with disturbance $d_t$

$$\begin{array}{rlr}{x}_{t+1}& =A{x}_t+B{u}_t& +A_d{d}_t\\ y_t& =C{x}_t& +C_d{d}_t\end{array}$$

Idea: Augment model

$$\begin{array}{rl}\left[\begin{array}{c}{x}_{t+1}\\ d_{t+1}\end{array}\right]& =\left[\begin{array}{cc}A& A_d\\ 0& \mathbb{I}\end{array}\right]\left[\begin{array}{c}{x}_t\\ d_t\end{array}\right]+\left[\begin{array}{c}B\\ 0\end{array}\right]u_t\\ y_t& =\left[\begin{array}{cc}C& C_d\end{array}\right]\left[\begin{array}{c}{x}_t\\ d_t\end{array}\right]\end{array}$$

Offset-free control

• Use state estimation with the equations above to estimate ${\hat{x}}_t$ and ${\hat{d}}_t$
• Use the equations above as model in MPC
• We require
  1. ($A_a,C_a$) to be observable
     • We can’t have state estimators on unobservable system
     • Means in practice that dim(${\hat{d}}_t$) $\le$ dim($y_t$)
  2. Target calculation must be modified to depend on ${\hat{d}}_t$.
     • Target calculation must be done at each time step.
  3. Typical industrial practice: $A_d=0$, $C_d=\mathbb{I}$. Called Bias Update
     • Aka disturbances only affect the output, and not the states ($A_d=0$)
     • Setting $C_d=\mathbb{I}$ makes it easier to determine the disturbance (since it is multiplied with $1$).
     • Does not require state estimators as a result.