lqg

Linear-Quadratic-Gaussian (LQG) design

Syntax

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(___,'current')
[reg,info] = lqg(___)


Description

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

$$J=E\left\{\underset{\tau \to \infty }{\lim }\frac{1}{\tau }{\displaystyle {\int }_0^{\tau }\left[x^T,u^T\right]Q_{xu}\left[\begin{array}{c}x\\ u\end{array}\right]dt}\right\}$$

subject to the plant equations

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

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

$$E\left(\left[\begin{array}{c}w\\ v\end{array}\right]\cdot \left[\begin{array}{cc}w\text{'}& v\text{'}\end{array}\right]\right)=QWV$$

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

$$J=E\left\{\underset{\tau \to \infty }{\lim }\frac{1}{\tau }{\displaystyle {\int }_0^{\tau }\left(\left[x^T,u^T\right]Q_{xu}\left[\begin{array}{c}x\\ u\end{array}\right]+x_i^T{Q}_i{x}_i\right)}dt\right\}$$

where x_i is the integral of the tracking error r - y. For MIMO systems, r, y, and x_i must have the same length.

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.

reg = lqg(___,'current') uses the "current" Kalman estimator, which uses x[n|n] as the state estimate when computing an LQG regulator for a discrete-time system.

[reg,info] = lqg(___) returns the controller and estimator gain matrices in the structure info for any of the previous syntaxes. You can use the controller and estimator gains to, for example, implement the controller in observer form. For more information, see Algorithms.

Examples

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

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

The system has the following noise covariance data:

$$\begin{array}{l}{Q}_n=E\left(w{w}^T\right)=\left[\begin{array}{ccc}4& 2& 0\\ 2& 1& 0\\ 0& 0& 1\end{array}\right]\\ R_n=E\left(v{v}^T\right)=0.7\end{array}$$

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

$$J(u)={\displaystyle {\int }_0^{\infty }\left(0.1x^{T}x+u^T\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]u\right)dt}$$

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

$$J(u)={\displaystyle {\int }_0^{\infty }\left(0.1x^{T}x+x_i^2+u^T\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]u\right)dt}$$

To design the LQG controllers for this system:

1. 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);

2. 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);

3. Form the LQG regulator by typing the following command:

   KLQG = lqg(sys,QXU,QWV)

   This command returns the following LQG regulator:

   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.

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

   KLQG1 = lqg(sys,QXU,QWV,QI,'1dof')

   This command returns the following LQG servo controller:

   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.

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

   KLQG2 = lqg(sys,QXU,QWV,QI,'2dof')

   This command returns the following LQG servo controller:

   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.

Tips

• lqg can be used for both continuous- and discrete-time plants. In discrete time, lqg uses x[n|n-1] as its state estimate by default. To use x[n|n] as the state estimate and compute the optimal LQG controller, use the 'current' input argument. For details on the state estimators, see kalman.

• To compute the LQG regulator, lqg uses the commands lqr and kalman. To compute the servo-controller, lqg uses the commands lqi and kalman.

• When you want more flexibility for designing regulators, you can use the lqr, kalman, and lqgreg commands. When you want more flexibility for designing servo controllers, you can use the lqi, kalman, and lqgtrack commands. For more information on using these commands and how to decide when to use them, see Linear-Quadratic-Gaussian (LQG) Design for Regulation and Linear-Quadratic-Gaussian (LQG) Design of Servo Controller with Integral Action.

Algorithms

The controller equations are:

• For continuous time:

  $$\begin{array}{c}d{x}_e=A{x}_e+Bu+L\left(y-C{x}_e-Du\right)\\ u=-K_x{x}_e-K_i{x}_i\end{array}$$

• For discrete time:

  $$x\left[n+1|n\right]=Ax\left[n|n-1\right]+Bu\left[n\right]+L\left(y\left[n\right]-Cx\left[n|n-1\right]-Du\left[n\right]\right)$$

  • Delayed estimator:

    $$u\left[n\right]=-K_{x}x\left[n|n-1\right]-K_i{x}_i\left[n\right]$$

  • Current estimator:

    $$\begin{array}{c}u\left[n\right]=-K_{x}x\left[n|n\right]-K_i{x}_i\left[n\right]-K_{w}w\left[n|n\right]\\ =-K_{x}x\left[n|n-1\right]-K_i{x}_i\left[n\right]-\left(K_x{M}_x+K_w{M}_w\right)y_{inn}\left[n\right]\end{array}$$

    $$y_{inn}\left[n\right]=y\left[n\right]-Cx\left[n|n-1\right]-Du\left[n\right]$$

Here,

• A, B, C, and D are the state-space matrices of the LQG regulator, reg.

• x_i is the integral of the tracking error r - y.

• K_x, K_w, K_i, L, M_x, and M_w are the controller and estimator gain matrices returned in info.

Version History

Introduced before R2006a