lqrpid(sys,Q,R,vara​rgin)

LQR-based output-feedback PID controller design for LTI ss systems

lqrpid: LQR-based PID output-feedback controller design for LTI ss systems.
[F,P,E] = lqrpid(sys,Q,R,N) calculates the optimal (if l=n/2) or the
sub-optimal (if l~=n/2) output-feedback PID (Proportional-Integral
-Derivative) gain matrix F = [Kp,Ki,Kd], such that:

* For a continuous-time state-space model SYS, the output-feedback
law:

u = - Kp y - Ki Integral y dt - Kd dy/dt,

minimizes the cost function:

J = Integral {x'Qx + u'Ru + 2x'Nu} dt

Subject to the system dynamics dx/dt = Ax + Bu; y = Cx

INPUTS:
REQUIRED
SYS - state-space LTI system (ss: sys.a,sys.b,sys.c...)
Q - weighting matrix related to states (Q >= 0 if N == 0)
R - positive definite weighting matrix (R > 0)

OPTIONAL
N - The matrix N is set to zero when omitted.
If N~=0 then Q-N*inv(R)*N'>=0 (use eig to check)

OUTPUTS:
F - static output-feedback gain matrix containing the
PID gains F = [Kp,Ki,Kd]
P - Lyapunov matrix
E - Closed-loop system eigen values

ASSUMPTIONS:
- The pair ([A,0;C,0],[B;0]) is stabilizable,
- R > 0 and Q-N*inv(R)*N' >= 0,
- Q-N*inv(R)*N' and A-B*inv(R)*N' have no unobservable mode on the
imaginary axis

OTHER INFO:
- The size of the weighting matrices is AUGMENTED:
Q(n+l,n+l), R(m,m), N(n+l,m)
where
n - number of states,
m - number of inputs,
l - number of outputs,
because the system is augmented with additional state variable(s)
for the PID controller design.
- Optimal solution is for l=n/2. If l~=n/2, an approximation is
calculated and tested for stability.

REQUIREMENTS:
Matlab:
- control system toolbox installed
Octave:
- control package installed and loaded

This function is based on:
S. Mukhopadhyay: P.I.D. equivalent of optimal regulator,
Electronics Letters, Vol. 14, No. 25, pp. 821-822, 1978.

Tested with Matlab 2014b/Octave 4.0