Multivariable control refers to controlling systems with more than one input or more than one actuator. Unlike single-input controllers, multivariable control methods such as LQR and MPC must handle interactions between actuators and states. This requires designing a control law that coordinates all actuators simultaneously rather than treating each one independently.

To form a mathematical rule (or law) that coordinates multiple actuators using multivariable control methods e.g. LQR or MPC, you need to follow a structured control-design process.

1. Start with a State-Space Model

For a system with multiple actuators, the dynamics are described by

\[\dot{x}(t) = Ax(t) + Bu(t)\]

or in discrete time

\[\dot{x}_{k+1} = Ax_{k} + Bu_{k}\]

Here, \(x\) contains all relevant states (positions, velocities, pressures, currents, etc.), and \(u\) contains the inputs for each actuator. Any coupling between actuators is captured naturally inside the matrices \(A\) and \(B\).

2. Define the Control Objective

LQR and MPC both use an optimization problem where the “goal” of the controller is encoded in a cost function. This is exactly where you define how actuators should work together.

For LQR, the cost is

\[J = \int_0^{\infty} (x^T Q x + u^T R u)\, dt\]

and for MPC

\[J = \sum_{k=0}^{N_p} (x_k^{T}Qx_k + u_k^{T}Ru_k)\]

3. The Mathematical Rule Between Actuators Is Encoded in Q and R

If you want actuators to behave similarly or follow a specific relation, you add penalties on their differences.

Eample rule:

Actuator 1 and actuator 2 should apply nearly the same input.

Add the term

\[(u_1 - u_2)^2\]

This creates a coupling in the R matrix and forces the controller to keep both actuators coordinated. The controller will automatically satisfy this rule while still optimizing performance.

4. LQR: Compute the Control Law

Solve the algebraic Riccati equation to obtain the feedback gain \(K\),

\[u = -Kx\]

The rows of \(K\) determine how each actuator reacts to all system states. If the rows are similar, the actuators naturally coordinate.

5. MPC: Optimization With Constraints

MPC minimizes the same style of cost function but also allows constraints. This means actuator relationships can be defined either as:

• Soft rules (penalties, like \((u_1 - u_2)^2\)). Soft rules influence behavior but do not strictly force it. They are encoded in Q and R.
• Hard constraints (e.g., \(u_1 = u_2\), or \(\vert u_1 - u_2 \vert \leq \epsilon\)).

MPC enforces these rules while predicting the future behavior of the system.

Summary

• The “mathematical rule” between actuators is created by how you structure the cost function.
• LQR encodes the rule inside the \(Q\) and \(R\) matrices, which shape the feedback gain \(K\).
• MPC enforces the rule through penalties and constraints during the optimization.
• Both methods require a state-space model of the system.