Designing an LQG Regulator

As an example of LQG design, consider the following regulation problem.

circuit model
Figure 1. Schematic of simple closed-loop regulator system.

The goal is to regulate the plant, the simple buck converter we have been using, around zero. The input disturbance d is low frequency with a given power spectral density PSD.

There is some measurement noise n, with noise intensity given by E(n²)=0.01.

Use the cost function

`J(u) = \int_0^\infty (10 y^2 + u^2) dt`

to specify the tradeoff between regulation performance and cost of control. The following equations represent our open-loop state-space model:

	dx/dt = Fo x + Go u + Go d   (state equations)
	 y    = Ho x + n             ( measurements)

where

	Fo = [  9.42e-001  1.90e-001
	       -1.90e-001  9.50e-001 ];
	Go = [  4.31e-001
	        3.03e+000 ];
	Ho = [  1.00e+000  0.00e+000 ];
	Ko = [  0.00e+000 ];
	Ts = 10e-6;

The following commands design the optimal LQG regulator F(s) for this problem:

	sys = ss(Fo,Go,Ho,Ko,Ts) % State-space plant model

	% Design LQ-optimal gain K
	K = lqry(sys,10,1)       % u = -Kx minimizes J(u)
	K = 
		1.1308    0.3317

	% Separate control input u and disturbance input d
	P = sys(:,[1 1]);        % input [u;d], output y

	% Design Kalman state estimator Kest.
	Kest = kalman(P,1,0.01)
 
	a =                    x1_e         x2_e
		 x1_e      -1.0789         0.19
		 x2_e      -5.5806         0.95

	b =                      u1           y1
		 x1_e        0.431       2.0209
		 x2_e         3.03       5.3906

	c =                    x1_e         x2_e
		 y1_e     0.037999            0
		 x1_e     0.037999            0
		 x2_e      -5.8668            1

	d =                      u1           y1
		 y1_e            0        0.962
		 x1_e            0        0.962
		 x2_e            0       5.8668

	Sampling time: 1e-005
	Discrete-time model.

With this information we can construct a Kalman filter and feedback matrix in LTSPICE as shown in Figure 2.

feedback circuit in LTSPICE
Figure 2. Schematic of simple closed-loop buck converter regulator system.

The results from this simple procedure, Fig. 3, are comparable to what we get with the non-automated procedure we showed before.

Step response plot from LTSPICE
Figure 3. Results for closed-loop buck converter regulator system.

The slight overshoot is caused by the fact that we are now not exactly in the steady-state where we computed our system matrices Fo, Go, Ho and Ko. If necessary, the response can be tuned by tweaking the coefficients (10 and 1) of J(u).

Summary

With the Kalman State Observer we can predict the inductor current instead of measuring it. In effect we are exchanging computing power for ADC and signal-conditioning hardware.