lqgtrack

Form Linear-Quadratic-Gaussian (LQG) servo controller

Syntax

C = lqgtrack(kest,k) C = lqgtrack(kest,k,'2dof') C = lqgtrack(kest,k,'1dof') C = lqgtrack(kest,k,...CONTROLS)

Description

lqgtrack forms a Linear-Quadratic-Gaussian (LQG) servo controller with integral action for the loop shown in the following figure. This compensator ensures that the output y tracks the reference command r and rejects process disturbances w and measurement noise v. lqgtrack assumes that r and y have the same length.

Note

Always use positive feedback to connect the LQG servo controller C to the plant output y.

C = lqgtrack(kest,k) forms a two-degree-of-freedom LQG servo controller C by connecting the Kalman estimator kest and the state-feedback gain k, as shown in the following figure. C has inputs $[r; y]$ and generates the command $u = - K [\hat{x}; x_{i}]$ , where $\hat{x}$ is the Kalman estimate of the plant state, and x_i is the integrator output.

The size of the gain matrix k determines the length of x_i. x_i, y, and r all have the same length.

The two-degree-of-freedom LQG servo controller state-space equations are

$\begin{array}{l} [\begin{matrix} \dot{\hat{x}} \\ {\dot{x}}_{i} \end{matrix}] = [\begin{matrix} A - B K_{x} - L C + L D K_{x} & - B K_{i} + L D K_{i} \\ 0 & 0 \end{matrix}] [\begin{matrix} \hat{x} \\ x_{i} \end{matrix}] + [\begin{matrix} 0 & L \\ I & - I \end{matrix}] [\begin{matrix} r \\ y \end{matrix}] \\ u = [\begin{matrix} - K_{x} & - K_{i} \end{matrix}] [\begin{matrix} \hat{x} \\ x_{i} \end{matrix}] \end{array}$

Note

The syntax C = lqgtrack(kest,k,'2dof') is equivalent to C = lqgtrack(kest,k).

C = lqgtrack(kest,k,'1dof') forms a one-degree-of-freedom LQG servo controller C that takes the tracking error e = r – y as input instead of [r ; y], as shown in the following figure.

The one-degree-of-freedom LQG servo controller state-space equations are

$\begin{array}{l} [\begin{matrix} \dot{\hat{x}} \\ {\dot{x}}_{i} \end{matrix}] = [\begin{matrix} A - B K_{x} - L C + L D K_{x} & - B K_{i} + L D K_{i} \\ 0 & 0 \end{matrix}] [\begin{matrix} \hat{x} \\ x_{i} \end{matrix}] + [\begin{matrix} - L \\ I \end{matrix}] e \\ u = [\begin{matrix} - K_{x} & - K_{i} \end{matrix}] [\begin{matrix} \hat{x} \\ x_{i} \end{matrix}] \end{array}$

C = lqgtrack(kest,k,...CONTROLS) forms an LQG servo controller C when the Kalman estimator kest has access to additional known (deterministic) commands U_d of the plant. In the index vector CONTROLS, specify which inputs of kest are the control channels u. The resulting compensator C has inputs

[U_d ; r ; y] in the two-degree-of-freedom case
[U_d ; e] in the one-degree-of-freedom case

The corresponding compensator structure for the two-degree-of-freedom cases appears in the following figure.

Examples

See the example Design an LQG Servo Controller.

Tips

You can use lqgtrack for both continuous- and discrete-time systems.

In discrete-time systems, integrators are based on forward Euler (see lqi for details). The state estimate $\hat{x}$ is either x[n|n] or x[n|n–1], depending on the type of estimator (see kalman for details).

For a discrete-time plant with equations:

$\begin{matrix} x [n + 1] = A x [n] + B u [n] + G w [n] \\ y [n] = C x [n] + D u [n] + H w [n] + v [n] {Measurements} \end{matrix}$

connecting the "current" Kalman estimator to the LQR gain is optimal only when $E (w [n] v {[n]}^{'}) = 0$ and y[n] does not depend on w[n] (H = 0). If these conditions are not satisfied, compute the optimal LQG controller using lqg.

Version History

Introduced in R2008b