integral LQR control: The LQR controller uses proportional state feedback. In some applications it
is customary to add integral feedback so the system output still goes to zero in the presence of step
disturbances. The LQR controller does not naturally have this integral action. To introduce integral
feedback into an LQR setting, we augment the system with a new state q(t) that represents the integral
of the system state that we want to go to zero. One then applies the LQR synthesis to this augmented
system. So consider the following system

x ̇ =
2
4
010
001
1
2 1
3
5 x(t) +
2
4
0
0
1
3
5 u(t) +
2
4
0
0
1
3
5 r(t)

where r(t) = ⇢ 1 t
0
0 otherwise is a step disturbance input.
user
Sure, I can explain how to add integral action to the LQR controller in this particular system. Here's a brief overview of the procedure:

1. Introduce a new state variable, q(t), which represents the integral of the state you want to make zero. In this case, we want to control the position of the system (x(t)), so q(t) would represent the integral of position.

2. Augment the original state-space model of the system by adding the new q(t) state. So, the augmented state vector would be [x(t), q(t)].

3. Define a new input, u_aug(t), where the term "aug" denotes that this is the input for the augmented model. U_aug(t) is chosen such that it drives both the original system (described by the first set of equations in your system model) and the new state q(t), i.e., q_dot(t) = u_aug(t).

4. Create a new output signal based on the second set of equations in your system model, i.e., y(t) = 2x(t) + 3u(t).

5. Define the performance index J to include the new state so that it is correct with the integral action property. This can be done through the additional integral-squared term i.e., J = 1/2 * (x_t(t)^2 + q(t)^2) + ...

6. Use the above to design the optimal LQR controller.

This method is known as "dynamic feedback compensator (DFC)" which can compensate the control system's nonconservativity. Typically, DFC contribute to better performance in nonlinear control systems. But, in this case, it will work correctly.
assistant
integral LQR control: The LQR controller uses proportional state feedback. In some applications it
is customary to add integral feedback so the system output still goes to zero in the presence of step
disturbances. The LQR controller does not naturally have this integral action. To introduce integral
feedback into an LQR setting, we augment the system with a new state q(t) that represents the integral
of the system state that we want to go to zero. One then applies the LQR synthesis to this augmented
system. So consider the following system

x ̇ =
2
4
010
001
1
2 1
3
5 x(t) +
2
4
0
0
1
3
5 u(t) +
2
4
0
0
1
3
5 r(t)

where r(t) = ⇢ 1 t
0
0 otherwise is a step disturbance input.
user
To design an integral LQR controller for the given system, we first need to augment the system with an integrator. The integrator state q(t) is defined as the integral of the error between the system output and the reference signal. Since the reference signal is zero in this case, the integrator state q(t) is the integral of the system output.

The augmented system can be written as:

[ ẋ(t) ] = [ A 0 ] [ x(t) ] + [ B ] [ u(t) ] + [ E ] [ r(t) ]
[ q̇(t) ] [ 0 I ] [ q(t) ] [ 0 ] [ 1 ]

where,
A = [ 2 4; 0 1; 0 0 ], B = [ 2; 4; 1 ], E = [ 0; 0; 1 ], and I is an identity matrix of appropriate dimensions.

The cost function to be minimized for the LQR design is:

J = ∫[ x(t)^T Q x(t) + u(t)^T R u(t) ] dt

where Q is the state weight matrix and R is the control weight matrix. The choice of Q and R depends on the design requirements.

To include integral action, we modify the cost function as:

J = ∫[ x(t)^T Q x(t) + u(t)^T R u(t) + q(t)^T P q(t) ] dt

where P is the weight matrix for the integrator state.

We then apply LQR synthesis to the augmented system using the modified cost function to obtain the optimal control gain matrix K. The control law is given by:

u(t) = - K [ x(t) ]
[ q(t) ]

where the first part of the gain matrix K corresponds to the proportional feedback, and the second part corresponds to the integral feedback. This controller will ensure that the system output goes to zero in the presence of step disturbances.

To summarize, the steps to design an integral LQR controller are:

1. Augment the system with an integrator.
2. Choose appropriate weight matrices Q, R, and P.
3. Apply LQR synthesis to the augmented system with the modified cost function.
4. Obtain the optimal control gain matrix K.
5. Design the control law using proportional and integral feedback.
assistant