control//adaptive control//MRAC
MRAC, model reference adaptive control, is a direct adaptive scheme that adjusts controller gains online so that the plant's closed-loop response imitates a chosen reference model, without knowing the plant's parameters; all it needs is the sign of the input gain, which way the actuator pushes. You first write down how you want the system to behave, the **reference model** (for example, vertical speed following its setpoint like a first-order system with a 0.5 s time constant), and the adaptation does the rest.
MRAC, model reference adaptive control, is a direct adaptive scheme that adjusts controller gains online so that the plant's closed-loop response imitates a chosen reference model, without knowing the plant's parameters; all it needs is the sign of the input gain, which way the actuator pushes. You first write down how you want the system to behave, the reference model (for example, vertical speed following its setpoint like a first-order system with a 0.5 s time constant), and the adaptation does the rest.
The scalar version holds the whole idea. Plant y˙=ay+bu\dot y=ay+buy˙=ay+bu with aaa and bbb unknown (bbb is, say, the inverse of the mass); reference model y˙m=−amym+amr\dot y_m=-a_my_m+a_mry˙m=−amym+amr; controller u=k^rr+k^yyu=\hat k_rr+\hat k_yyu=k^rr+k^yy. The adaptation law is
k^˙r=−γ sgn(b) e r,k^˙y=−γ sgn(b) e y,e=y−ym.\dot{\hat k}_r=-\gamma\,\operatorname{sgn}(b)\,e\,r,\qquad \dot{\hat k}_y=-\gamma\,\operatorname{sgn}(b)\,e\,y,\qquad e=y-y_m.k^˙r=−γsgn(b)er,k^˙y=−γsgn(b)ey,e=y−ym.
Read it as: each gain is corrected in proportion to the model error times the signal that gain multiplies. If the plant runs above the model while the setpoint is positive, k^r\hat k_rk^r goes down. It is a continuous-time gradient step on e2/2e^2/2e2/2, and γ>0\gamma>0γ>0 is the adaptation gain. Why it does not run away is a Lyapunov argument: the energy-like V=12e2+∣b∣2γ(k~r2+k~y2)V=\tfrac12e^2+\tfrac{|b|}{2\gamma}(\tilde k_r^2+\tilde k_y^2)V=21e2+2γ∣b∣(k~r2+k~y2), mixing tracking error and gain error, has V˙=−ame2≤0\dot V=-a_me^2\le0V˙=−ame2≤0, so everything stays bounded and eee tends to zero.
MRAC RMS error0.2 cm fixed RMS error12.3 cm estimated mass2.50 kg true mass2.5 kg A 2.5 kg mass follows a square setpoint under two controllers of the same form. After 20 s the adaptive one (γ 2.00) has estimated the mass at 2.50 kg and stays 0.2 cm RMS from the reference model; the fixed one, tuned for 1 kg, misses it by 12.3 cm.
Press Add load twice and watch the adaptive controller pull its estimated mass up to the true one while the fixed controller keeps missing; then push γ\gammaγ to a few hundred and the estimate turns nervous, or down to 0.05 and it takes many cycles.
The adaptation gain is the dangerous dial.
Too low and the adapter is nearly as clumsy as a fixed controller; too high and the command and the estimate turn nervous, and with unmodelled motor lag, noise or disturbances the parameters can drift and burst (robust adaptation).
The Lyapunov proof says the tracking error vanishes, not that the gains converge to the ideal ones. Without persistent excitation tracking can be perfect with wrong parameters; in the figure the estimate moves only at setpoint changes.
Error times signal is one rule found three times: the MRAC law, the LMS of adaptive filters and stochastic gradient descent, and the Kalman correction (gain times innovation) all adjust numbers until a prediction stops being wrong.
Maturity: niche, in concrete applications such as thrust estimators; most adaptation in production is gain scheduling or a self-tuning estimate of one parameter (adaptive control).