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Linear-Quadratic-Gaussian (LQG) design

`reg = lqg(sys,QXU,QWV)`

reg = lqg(sys,QXU,QWV,QI)

reg = lqg(sys,QXU,QWV,QI,'1dof')

reg = lqg(sys,QXU,QWV,QI,'2dof')

`reg = lqg(sys,QXU,QWV)`

computes an optimal
linear-quadratic-Gaussian (LQG) regulator `reg`

given
a state-space model `sys`

of the plant and weighting
matrices `QXU`

and `QWV`

. The dynamic
regulator `reg`

uses the measurements *y* to
generate a control signal *u* that regulates *y* around
the zero value. Use positive feedback to connect this regulator to
the plant output *y*.

The LQG regulator minimizes the cost function

$$J=E\left\{\underset{\tau \to \infty}{\mathrm{lim}}\frac{1}{\tau}{\displaystyle {\int}_{0}^{\tau}\left[{x}^{T},{u}^{T}\right]{Q}_{xu}\left[\begin{array}{c}x\\ u\end{array}\right]dt}\right\}$$

subject to the plant equations

dx/dt = Ax + Bu + w y = Cx + Du + v

where the process noise *w* and measurement
noise *v* are Gaussian white noises with covariance:

E([w;v] * [w',v']) = QWV

`reg = lqg(sys,QXU,QWV,QI)`

uses the setpoint
command *r* and measurements *y* to
generate the control signal *u*. `reg`

has
integral action to ensure that *y* tracks the command *r*.

The LQG servo-controller minimizes the cost function

$$J=E\left\{\underset{\tau \to \infty}{\mathrm{lim}}\frac{1}{\tau}{\displaystyle {\int}_{0}^{\tau}\left(\left[{x}^{T},{u}^{T}\right]{Q}_{xu}\left[\begin{array}{c}x\\ u\end{array}\right]+{x}_{i}^{T}{Q}_{i}{x}_{i}\right)}\text{\hspace{0.05em}}dt\right\}$$

where *x _{i}* is
the integral of the tracking error

`reg = lqg(sys,QXU,QWV,QI,'1dof')`

computes
a one-degree-of-freedom servo controller that takes *e* = *r* - *y* rather
than [*r* ; *y*] as input.

`reg = lqg(sys,QXU,QWV,QI,'2dof')`

is equivalent
to `LQG(sys,QXU,QWV,QI)`

and produces the two-degree-of-freedom
servo-controller shown previously.

**Linear-Quadratic-Gaussian (LQG) Regulator
and Servo Controller Design**

This example shows how to design an linear-quadratic-Gaussian (LQG) regulator, a one-degree-of-freedom LQG servo controller, and a two-degree-of-freedom LQG servo controller for the following system.

The plant has three states
(*x*), two control inputs (*u*),
three random inputs (*w*), one output (*y*),
measurement noise for the output (*v*), and the following
state and measurement equations.

$$\begin{array}{l}\frac{dx}{dt}=Ax+Bu+w\\ y=Cx+Du+v\end{array}$$

where

$$\begin{array}{l}A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]\text{\hspace{1em}}\text{\hspace{1em}}B=\left[\begin{array}{cc}0.3& 1\\ 0& 1\\ -0.3& 0.9\end{array}\right]\\ \\ C=\left[\begin{array}{ccc}1.9& 1.3& 1\end{array}\right]\text{\hspace{1em}}D=\left[\begin{array}{cc}0.53& -0.61\end{array}\right]\end{array}$$

The system has the following noise covariance data:

$$\begin{array}{l}{Q}_{n}=E\left(\omega {\omega}^{T}\right)=\left[\begin{array}{ccc}4& 2& 0\\ 2& 1& 0\\ 0& 0& 1\end{array}\right]\\ {R}_{n}=E\left(v{v}^{T}\right)=0.7\end{array}$$

For the regulator, use the following cost function to define the tradeoff between regulation performance and control effort:

$$J(u)={\displaystyle {\int}_{0}^{\infty}\left(0.1{x}^{T}x+{u}^{T}\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]u\right)dt}$$

For the servo controllers, use the following cost function to define the tradeoff between tracker performance and control effort:

$$J(u)={\displaystyle {\int}_{0}^{\infty}\left(0.1{x}^{T}x+{x}_{i}^{2}+{u}^{T}\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]u\right)dt}$$

To design the LQG controllers for this system:

Create the state-space system by typing the following in the MATLAB Command Window:

A = [0 1 0;0 0 1;1 0 0]; B = [0.3 1;0 1;-0.3 0.9]; C = [1.9 1.3 1]; D = [0.53 -0.61]; sys = ss(A,B,C,D);

Define the noise covariance data and the weighting matrices by typing the following commands:

nx = 3; %Number of states ny = 1; %Number of outputs Qn = [4 2 0; 2 1 0; 0 0 1]; Rn = 0.7; R = [1 0;0 2] QXU = blkdiag(0.1*eye(nx),R); QWV = blkdiag(Qn,Rn); QI = eye(ny);

Form the LQG regulator by typing the following command:

This command returns the following LQG regulator:KLQG = lqg(sys,QXU,QWV)

a = x1_e x2_e x3_e x1_e -6.212 -3.814 -4.136 x2_e -4.038 -3.196 -1.791 x3_e -1.418 -1.973 -1.766 b = y1 x1_e 2.365 x2_e 1.432 x3_e 0.7684 c = x1_e x2_e x3_e u1 -0.02904 0.0008272 0.0303 u2 -0.7147 -0.7115 -0.7132 d = y1 u1 0 u2 0 Input groups: Name Channels Measurement 1 Output groups: Name Channels Controls 1,2 Continuous-time model.

Form the one-degree-of-freedom LQG servo controller by typing the following command:

This command returns the following LQG servo controller:KLQG1 = lqg(sys,QXU,QWV,QI,'1dof')

a = x1_e x2_e x3_e xi1 x1_e -7.626 -5.068 -4.891 0.9018 x2_e -5.108 -4.146 -2.362 0.6762 x3_e -2.121 -2.604 -2.141 0.4088 xi1 0 0 0 0 b = e1 x1_e -2.365 x2_e -1.432 x3_e -0.7684 xi1 1 c = x1_e x2_e x3_e xi1 u1 -0.5388 -0.4173 -0.2481 0.5578 u2 -1.492 -1.388 -1.131 0.5869 d = e1 u1 0 u2 0 Input groups: Name Channels Error 1 Output groups: Name Channels Controls 1,2 Continuous-time model.

Form the two-degree-of-freedom LQG servo controller by typing the following command:

This command returns the following LQG servo controller:KLQG2 = lqg(sys,QXU,QWV,QI,'2dof')

a = x1_e x2_e x3_e xi1 x1_e -7.626 -5.068 -4.891 0.9018 x2_e -5.108 -4.146 -2.362 0.6762 x3_e -2.121 -2.604 -2.141 0.4088 xi1 0 0 0 0 b = r1 y1 x1_e 0 2.365 x2_e 0 1.432 x3_e 0 0.7684 xi1 1 -1 c = x1_e x2_e x3_e xi1 u1 -0.5388 -0.4173 -0.2481 0.5578 u2 -1.492 -1.388 -1.131 0.5869 d = r1 y1 u1 0 0 u2 0 0 Input groups: Name Channels Setpoint 1 Measurement 2 Output groups: Name Channels Controls 1,2 Continuous-time model.

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