Routine iomargins

Compute bounds on the worst-case gain, phase, modulus and delay margins with desired accuracy.

Description

Let us consider the following interconnection between a nominal LTI system $M(s)$ and a block-diagonal operator $\Delta(s)$.

Image

Lower and/or upper bounds $M_{lb}$ and $M_{ub}$ are computed on the highest value $M_{max}$ of the (real or complex) gain, the phase shift or the time-delay that can be inserted between $y$ and $e$ without destabilizing the interconnection when $\Delta(s)$ takes any admissible value (see problem 3 on the overview page of the toolbox or more details). More precisely, some additional fictitious uncertainties $\delta_{M}=diag(\delta_{M,i})$ corresponding to gain, phase, modulus or time-delay variations are connected between $y$ and $e$ (see also the example at the bottom of this page):

• $\delta_{M,i}=1+\hat{\delta}_i$, where $\hat{\delta}_i$ is real, if the worst-case gain margin is to be evaluated,
• $\delta_{M,i}=e^{j\hat{\delta}_i}$, where $\hat{\delta}_i$ is real, if the worst-case phase margin is to be evaluated,
• $\delta_{M,i}=1+\hat{\delta}_i$, where $\hat{\delta}_i$ is complex, if the worst-case modulus margin is to be evaluated,
• $\delta_{M,i}=e^{-\hat{\delta}_is}$, where $\hat{\delta}_i$ is positive real, if the worst-case delay margin is to be evaluated.

In this context, the value of $M_{max}$ corresponds to the maximum value of $|\hat{\delta_i}|$ for which the interconnection is stable for any admissible value of $\Delta(s)$. The lower bound $M_{lb}$ is guaranteed on a whole frequency interval. Moreover, a branch and bound algorithm can be applied, in which case the bounds satisfy $M_{ub}\le (1+\epsilon)M_{lb}$, where $\epsilon$ is a user-defined threshold. A value $\tilde{\Delta}(s)$ of $\Delta(s)$ for which the upper bound $M_{ub}$ is obtained is also determined, and the considered stability region can be bounded either by the imaginary axis or by a truncated sector. Note: The initial interconnection is normalized with convert_data before the problem is solved.

Syntax

[bnds,wc,pert,iodesc]=iomargins(usys,margin{,options}) [bnds,wc,pert,iodesc]=iomargins(gsys,margin{,options}) [bnds,wc,pert,iodesc]=iomargins(lsys,margin{,options}) [bnds,wc,pert,iodesc]=iomargins(sys,blk,margin{,options})

Input arguments

The interconnection between $M(s)$ and $\Delta(s)$ can be described by:

• a USS object usys,
• a GSS object gsys,
• a LFR object lsys,
• a LTI object sys describing the LTI system $M(s)$ and a $N\times 2$ matrix blk defining the structure of the block-diagonal operator $\Delta(s)=diag\left(\Delta_1(s),\dots,\Delta_N(s)\right)$:
  • blk(i,:)=[-ni 0] $\Rightarrow$ $\Delta_i(s)=\delta_iI_{n_i}$ with $\delta_i$ real,
  • blk(i,:)=[ni 0] $\Rightarrow$ $\Delta_i(s)=\delta_iI_{n_i}$ with $\delta_i$ complex,
  • blk(i,:)=[ni mi] $\Rightarrow$ $\Delta_i(s)$ is a $n_i\times m_i$ LTI system.

The second input argument margin is a character array which allows to specify the considered problem:

• it must contain exactly one of the following characters : 'g' for gain margin, 'p' for phase margin, 'm' for modulus margin or 'd' for delay margin,
• it must also contain 'l', 'u', 'lu' or 'b' depending on whether only a lower bound, only an upper bound, both lower and upper bounds, or both lower and upper bounds with desired accuracy have to be computed.

The last input argument options is an optional structured variable with fields:

Additional fields can be defined (see muub_mixed for lower bound computation, mulb_mixed or mulb_nreal for upper bound computation and mubb_mixed for branch and bound), but the ones listed above are usually sufficient for non-expert users.

Output arguments

Example

Let us consider the longitudinal missile model depicted in the figure below. The open-loop model has two states (angle of attack and pitch rate), one input (tail fin deflection $\delta$) and two outputs (acceleration $\eta$ and pitch rate $q$). Four parametric uncertainties on the stability derivatives are considered. The open-loop model can thus be written as an LFR, where $\Delta=diag(\delta_1,\delta_2,\delta_3,\delta_4)$. Each of the $\delta_i$ is normalized and the corresponding stability derivative varies by $\pm 50\%$ when $\delta_i\in [-1,1]$. The actuator is modeled by a second order system with frequency $\omega_a=$150 rad/s and damping ratio $\xi_a=$0.7. The controller has 10 states and is designed by an $\mathcal{H}_\infty$ method. Thus the whole plant has 14 states and 4 non-repeated parametric uncertainties.

Image

The objective is to compute reliable estimates of the worst-case gain, phase, modulus and delay margins at the output of the plant when $\Delta$ takes all possible values in the unit ball of admissible perturbations. Two additional inputs $\tilde{\eta},\tilde{q}$ and outputs $\eta,q$ are thus created, and the missile model sys to be used with iomargins is depicted in black in the figure above.
load missile_model
minfo(sys)
blk=[-1 0;-1 0;-1 0;-1 0]
The first 4 inputs $w$ and outputs $z$ of sys correspond to the parametric uncertainties $\delta_1$ to $\delta_4$, while the last 2 inputs $\tilde{\eta},\tilde{q}$ and outputs $\eta,q$ are used for worst-case margins computation. Note that the $\delta_{M,i}$ are not included in blk.

Optional arguments for lower bound computation:
options.lmi=[0.001 100 1e10 5 1];
options.tol=0;

1. Worst-case gain margin

Let $\delta_{M,i}=1+\hat{\delta}_i$, where $\hat{\delta}_i$ is real. The worst-case gain margin is the highest value of $M$ such that the interconnection is stable for all $\Delta\in\mathcal{B}_{{\bf\Delta}}$ and for all $|\hat{\delta}_i| < M$.
Computation of lower and upper bounds:
[bnds,wc,pert,iodesc]=iomargins(sys,blk,'glu',options);
(bnds(2)-bnds(1))/bnds(1)
The gap between the bounds is 0.01%.
D=mdiag(pert,eye(2)+diag(iodesc{5}.value));
damp(feedback(sys,D,1))
Two eigenvalues of the interconnection lie on the imaginary axis.

2. Worst-case phase margin

Let $\delta_{M,i}=e^{j\hat{\delta}_i}$, where $\hat{\delta}_i$ is real. The worst-case gain margin is the highest value of $M$ such that the interconnection is stable for all $\Delta\in\mathcal{B}_{{\bf\Delta}}$ and for all $|\hat{\delta}_i| < M$.
Computation of lower and upper bounds: 
bnds,wc,pert,iodesc]=iomargins(sys,blk,'plu',options);
(bnds(2)-bnds(1))/bnds(1)
The gap between the bounds is 0.04%.
D=mdiag(pert,diag(exp(sqrt(-1)*iodesc{5}.value)));
damp(feedback(sys,D,1))
One eigenvalue of the interconnection lies on the imaginary axis.

3. Worst-case modulus margin

Let $\delta_{M,i}=1+\hat{\delta}_i$, where $\hat{\delta}_i$ is complex. The worst-case modulus margin is the highest value of $M$ such that the eigenvalues of the interconnection are inside the considered sector for all $\Delta\in\mathcal{B}_{{\bf\Delta}}$ and for all $|\hat{\delta}_i| < M$.
options.sector=[-0.3 0.2];
Computation of lower and upper bounds:
[bnds1,wc,pert,iodesc]=iomargins(sys,blk,'mlu',options);
(bnds1(2)-bnds1(1))/bnds1(1)
The gap between the bounds is 0.0002%.
D=mdiag(pert,eye(2)+diag(iodesc{5}.value)); display_sector(options.sector,feedback(sys,D,1));
One eigenvalue of the interconnection lies on the boundary of the considered sector. options=rmfield(options,'sector');

4. Worst-case delay margin

Let $\delta_{M,i}=e^{-\hat{\delta}_is}$, where $\hat{\delta}_i$ is positive real. The worst-case delay margin is the highest value of $M$ such that the interconnection is stable for all $\Delta\in\mathcal{B}_{{\bf\Delta}}$ and for all $0\le \hat{\delta}_i < M$.
Computation of lower and upper bounds:
options.grid=-20;
[bnds1,wc,pert,iodesc]=iomargins(sys,blk,'dlu',options);
(bnds1(2)-bnds1(1))/bnds1(1)
The gap between the bounds is 1%.
s=tf('s');
D=mdiag(pert,exp(-iodesc{5}.value(1)*s),exp(-iodesc{5}.value(2)*s));damp(pade(feedback(sys,D,1),5))
One eigenvalue of the interconnection lies on the imaginary axis.

See also

muub_mixed
mulb_mixed
mulb_nreal
hinflb_real
mubb_mixed
convert_data
make_square

References