[K,CL,GAM,INFO]=loopsyn(G,Gd) [K,CL,GAM,INFO]=loopsyn(G,Gd,RANGE)
loopsyn
is an H_{∞} optimal
method for loopshaping control synthesis. It computes a stabilizing H_{∞}controller K for
plant G to shape the sigma
plot
of the loop transfer function GK to have desired
loop shape G_{d} with accuracy γ = GAM
in
the sense that if ω_{0} is
the 0 db crossover frequency of the sigma
plot
of G_{d}(jω),
then, roughly,
$$\underset{\xaf}{\sigma}\left(G(j\omega )K(j\omega )\right)\ge \frac{1}{\gamma}\text{}\underset{\xaf}{\sigma}\left({G}_{d}(j\omega )\right)\text{forall}\omega {\omega}_{0}$$ | (1-14) |
$$\underset{\xaf}{\sigma}\left(G(j\omega )K(j\omega )\right)\le \gamma \text{}\underset{\xaf}{\sigma}\left({G}_{d}(j\omega )\right)\text{forall}\omega {\omega}_{0}$$ | (1-15) |
The STRUCT array INFO
returns additional
design information, including a MIMO stable min-phase shaping pre-filter W,
the shaped plant G_{s} = GW,
the controller for the shaped plant K_{s} = WK,
as well as the frequency range {ω_{min},ω_{max}}
over which the loop shaping is achieved
Input Argument | Description |
---|---|
G | LTI plant |
Gd | Desired loop-shape (LTI model) |
RANGE | (optional, default |
Output Argument | Description |
---|---|
K | LTI controller |
CL= G*K/(I+GK) | LTI closed-loop system |
GAM | Loop-shaping accuracy ( |
INFO | Additional output information |
INFO.W | LTI pre-filter W satisfying σ(G_{d}) = σ (GW) for all ω; W is always minimum-phase. |
INFO.Gs | LTI shaped plant: G_{s} = GW. |
INFO.Ks | LTI controller for the shaped plant: K_{s} = WK. |
INFO.range | {ω_{min},ω_{max}}
cell-array containing the approximate frequency range over which loop-shaping
could be accurately achieved to with accuracy |
The plant G must be stabilizable and detectable, must have at least as many inputs as outputs, and must be full rank; i.e,
size(G,2)
≥ size(G,1)
rank(freqresp(G,w)) = size(G,1)
for
some frequency w.
The order of the controller K can be large. Generically, when G_{d} is given as a SISO LTI, then the order N_{K} of the controller K satisfies
N_{K} = N_{Gs} + N_{W}
= N_{y}N_{Gd} + N_{RHP} + N_{W}
= N_{y}N_{Gd} + N_{RHP} + N_{G}
where
N_{y} denotes the number of outputs of the plant G.
N_{RHP} denotes the total number of nonstable poles and nonminimum-phase zeros of the plant G, including those on the stability boundary and at infinity.
N_{G}, N_{Gs}, N_{Gd} and N_{W} denote the respective orders of G, G_{s}, G_{d} and W.
Model reduction can help reduce the order of K —
see reduce
and ncfmr
.
[1] Le, V.X., and M.G. Safonov. Rational matrix GCD's and the design of squaring-down compensators—a state space theory. IEEE Trans. Autom.Control, AC-36(3):384–392, March 1992.
[2] Glover, K., and D. McFarlane. Robust stabilization of normalized coprime factor plant descriptions with H_{∞}-bounded uncertainty. IEEE Trans. Autom. Control, AC-34(8):821–830, August 1992.
[3] Chiang, R.Y., and M.G. Safonov. H_{∞} synthesis using a bilinear pole-shifting transform. AIAA J. Guidance, Control and Dynamics, 15(5):1111–1115, September–October 1992.