ncfmargin

Calculate normalized coprime stability margin of plant-controller feedback loop

Syntax

[marg,freq] = ncfmargin(P,C)

[marg,freq] = ncfmargin(P,C,sign)

[marg,freq] = ncfmargin(___,tol)

Description

[marg,freq] = ncfmargin(P,C) returns the normalized coprime stability margin of the multivariable feedback loop consisting of a controller C in negative feedback with a plant P:

The normalized coprime robust stability margin (also called the gap metric stability margin) is an indication of robustness to unstructured perturbations. Values greater than 0.3 generally indicate good robustness margins.

[marg,freq] = ncfmargin(P,C,sign) specifies the sign of the feedback connection assumed for the margin calculation. By default, sign = -1. Set sign = +1 for positive-feedback interconnection.

[marg,freq] = ncfmargin(___,tol) calculates the normalized coprime factor metric with the specified relative accuracy.

Examples

Normalized Coprime Stability Margin

Consider an unstable first-order plant, p, stabilized by high-gain and low-gain controllers, cL and cH.

p = tf(4,[1 -0.001]); 	
cL = 1;				
cH = 10;

Compute the stability margin of the closed-loop system with the low-gain controller.

[margL,~] = ncfmargin(p,cL)

margL = 
0.7069

Similarly, compute the stability margin of the closed-loop system with the high-gain controller.

[margH,~] = ncfmargin(p,cH)

margH = 
0.0995

The closed-loop systems with low-gain and high-gain controllers have normalized coprime stability margins of about 0.71 and 0.1, respectively. This result indicates that the closed-loop system with low-gain controller is more robust to unstructured perturbations than the system with the high-gain controller.

To observe this difference in robustness, construct an uncertain plant, punc, that has additional unmodeled dynamics at high frequency compared to the nominal plant.

punc = p + ultidyn('uncstruc',[1 1],'Bound',1);
sigma(p,punc,'r--')

Calculate the robust stability of the closed-loop systems formed by the uncertain plant and each controller.

[stabmargL,~] = robstab(feedback(punc,cL))

stabmargL = struct with fields:
           LowerBound: 0.9980
           UpperBound: 1
    CriticalFrequency: Inf

[stabmargH,~] = robstab(feedback(punc,cH))

stabmargH = struct with fields:
           LowerBound: 0.0998
           UpperBound: 0.1000
    CriticalFrequency: Inf

As expected, the robust stability analysis shows that the closed-loop system with low-gain controller is more robustly stable in the presence of the unmodeled LTI dynamics. In fact, this closed-loop system can tolerate almost 100% of the specified uncertainty. In contrast, closed-loop system with the high-gain controller can tolerate only about 10% of the specified uncertainty.

Compute Gap Metric and Stability Margin

Consider a plant and a stabilizing controller.

P1 = tf([1 2],[1 5 10]);
C = tf(4.4,[1 0]);

Compute the stability margin for this plant and controller.

b1 = ncfmargin(P1,C)

b1 = 
0.1961

Next, compute the gap between P1 and the perturbed plant, P2.

P2 = tf([1 1],[1 3 10]);
[gap,nugap] = gapmetric(P1,P2)

gap = 
0.1391

nugap = 
0.1390

Because the stability margin b1 = b(P1,C) is greater than the gap between the two plants, C also stabilizes P2. As discussed in Gap Metrics and Stability Margins, the stability margin b2 = b(P2,C) satisfies the inequality asin(b(P2,C)) ≥ asin(b1)-asin(gap). Confirm this result.

b2 = ncfmargin(P2,C);
[asin(b2) asin(b1)-asin(gap)]

ans = 1×2

    0.0997    0.0579

Input Arguments

P — Plant

dynamic system model

Plant, specified as a dynamic system model. P can be SISO or MIMO, as long as P*C has the same number of inputs and outputs. P can be continuous time or discrete time. If P is a generalized state-space model (genss or uss) then ncfmargin uses the current or nominal value of all control design blocks in P.

C — Controller

dynamic system model

Plant, specified as a dynamic system model. C can be SISO or MIMO, as long as P*C has the same number of inputs and outputs. C can be continuous time or discrete time. If C is a generalized state-space model (genss or uss) then ncfmargin uses the current or nominal value of all control design blocks in P.

By default, ncfmargin assumes a negative-feedback interconnection between P and C. To compute the margins for a closed-loop system with positive feedback, use [marg,freq] = ncfmargin(P,C,+1).

sign — Sign of feedback

-1 (default) | +1

Sign of the feedback connection, specified as either -1 or +1.

The default value, sign = -1, specifies negative feedback. Setting sign = +1 assumes a positive feedback connection for the margin calculation, as in the following diagram.

tol — Relative accuracy

0.001 (default) | positive scalar less than 1

Relative accuracy for the computed margin, specified as a positive scalar value less than 1. The default value is 0.001, or 0.1% accuracy.

Output Arguments

marg — Normalized coprime robust stability margin

scalar in [0,1]

Normalized coprime robust stability margin, returned as a scalar in the range [0,1]. This quantity, also known as the gap metric stability margin, is an indicator of closed-loop robustness to unstructured perturbations. For negative-feedback control architecture, It is defined as:

$$b(P,C)={\Vert \left[\begin{array}{c}I\\ C\end{array}\right]{(I+PC)}^{-1}\left[\begin{array}{cc}I& P\end{array}\right]\Vert }_{\infty }^{-1}={\Vert \left[\begin{array}{c}I\\ P\end{array}\right]{(I+CP)}^{-1}\left[\begin{array}{cc}I& C\end{array}\right]\Vert }_{\infty }^{-1}.$$

Values greater than 0.3 generally indicate good robustness margins. If the closed-loop system is unstable, then marg = 0. The quantity b(P,C)^–1 is the signal gain from disturbances on the plant input and output to the input and output of the controller.

freq — Frequency at which margin occurs

scalar | NaN

Frequency at which the margin marg occurs, returned as a scalar. If the closed-loop system is unstable, then freq = NaN.

More About

Stability Margin and Gap Metrics

The stability margin b(P,C) is related to the gap metric, which gives a numerical value δ(P₁,P₂) for the distance between two LTI systems (see gapmetric).

For both the gap and ν-gap metrics, the following robust performance result holds:

arcsin b(P₂,C₂) ≥ arcsin b(P₁,C₁) – arcsin δ(P₁,P₂) – arcsin δ(C₁,C₂),

To interpret this result, suppose that a nominal plant P₁ is stabilized by controller C₁ with stability margin b(P₁,C₁). Then, if P₁ is perturbed to P₂ and C₁ is perturbed to C₂, the stability margin is degraded by no more than the above formula.

The ν-gap is always less than or equal to the gap, so its predictions using the above robustness result are tighter.

Tips

• While ncfmargin assumes a negative-feedback loop, the ncfsyn command designs a controller for a positive-feedback loop. Therefore, to compute the margin using a controller designed with ncfsyn, use [marg,freq] = ncfmargin(P,C,+1).

Algorithms

The computation of the normalized coprime stability margin is as described in Chapter 16 of [1].

References

[1] Zhou, K., Doyle, J.C., Essentials of Robust Control. London, UK: Pearson, 1997.

Version History

Introduced before R2006a