control//feedback control
How to use inputs and measurements to modify an evolution without confusing plant and controller. The **plant** is the process that receives the actuation. The **controller** computes that actuation. The **closed-loop dynamics** belongs to their interconnection.
How to use inputs and measurements to modify an evolution without confusing plant and controller. The plant is the process that receives the actuation. The controller computes that actuation. The closed-loop dynamics belongs to their interconnection.
For a linear plant x˙=Fx+Bu\dot x=Fx+Bux˙=Fx+Bu, y=Cxy=Cxy=Cx, a state feedback u=−Kx+Nru=-Kx+Nru=−Kx+Nr produces
x˙=(F−BK)x+BNr.\dot x=(F-BK)x+BNr.x˙=(F−BK)x+BNr.
KKK and NNN are design choices. The eigenvalues of FFF describe the open plant; those of F−BKF-BKF−BK the ideal closed loop. They must not be attributed indistinctly to "the system" without naming the configuration.
Different goals. Stabilizing means making a desired equilibrium stable. Regulating means holding an output at a constant reference. Tracking a varying reference means controlling the error e=r−ye=r-ye=r−y, for example so that it tends to zero. Stability does not automatically imply zero steady-state error or a fast transient, and proves nothing about robustness to uncertainty, delays, noise or actuator limits.
A vertical drone. With mq¨=T−mgm\ddot q=T-mgmq¨=T−mg and a constant reference, one proposal is
T=mg−kp(q−r)−kdq˙.T=mg-k_p(q-r)-k_d\dot q.T=mg−kp(q−r)−kdq˙.
mgmgmg nominally cancels the weight; the other terms correct position and velocity. The rule does not become a physical law: it orders an actuation the motor will have to deliver. For a twice-differentiable reference, with known state and an ideal actuator,
T=m(g+r¨)−kp(q−r)−kd(q˙−r˙)T=m(g+\ddot r)-k_p(q-r)-k_d(\dot q-\dot r)T=m(g+r¨)−kp(q−r)−kd(q˙−r˙)
gives an error satisfying me¨+kde˙+kpe=0m\ddot e+k_d\dot e+k_pe=0me¨+kde˙+kpe=0, taking e=q−re=q-re=q−r here. The error-sign convention is declared before interpreting signs. The full case is vertical drone.
Observing is not measuring everything. A linear observer can take
x^˙=Fx^+Bu+Lo(y−Cx^).\dot{\hat x}=F\hat x+Bu+L_o(y-C\hat x).x^˙=Fx^+Bu+Lo(y−Cx^).
The innovation y−Cx^y-C\hat xy−Cx^ corrects the estimate. The observer needs suitable observability or detectability (controllability and observability); it does not invent missing information.
A PID adds proportional reaction, integral memory and an action based on the derivative of the error. The controller's memory is part of the total state. Motor lags, saturations and filters also change the loop that has to be analyzed: see feedback loop for the interconnection and delay and lag for the timing limits. A Lyapunov function can supply guarantees under explicit hypotheses.