What is about the plant? Assuming you can show boundedness of the error for a closed-loop system with each controller separately, you should be able to provide stability analysis of the combined system as well. If the closed-loop system is linear, it is pretty simple to analyze the poles. Consider the following extremely simplified one-dimensional example:

Dx= a*x + b*u,

y = x,

where the plant itself (w/o b*u-part) is ustable, a > 0. You can stabilize it by the feedback control u = -k*y, s.t. k is positive and a - k < 0. Now suppose you have a weighted sum of such feedback control laws instead:  u = -w1*k1*y - w2*k2*y - w3*k3*y, where the weights sum up to one. The stability condition of the closed-loop system now amounts to a - w1*k1 - w2*k2 - w3*k3  < 0 assuming each k satisfies: a - k < 0. For which w's does this condition hold? Suppose, k1 < k2 < k3. Then, w1*k1 + w2*k2 + w3*k3 > w1*k1 + w2*k1 + w3*k1 =  k1*(w1 + w2 + w3)  = k1 > a and you see that is does not depend on weights.

Addition: you can also interpret weights as baricentric coordinates in a convex hull generated by the control gains. Evidently, each point defined this way lies inside of the convex hull.

Thanks Pavel  - but what if the weights are time varying? I guess this again comes down to the analysis of the sub-filters operating by themselves where the gain is time varying between 0 and 1. How would I do this? I guess the varying gain could be considered to be a modulating function, so this would be a convolution in the complex frequency (s or z) domain. Is analysis possible if I don't know in advance how the weight will vary with time?    

It is a rather well-known result that the LPV model resulting from the mix of several LTI models is not necessarily stable even if the LTI models obtained with fixed values of the parameters are stable, as soon as the scheduling parameters are allowed to vary. You might refer to the survey paper by Rugh and Shamma on gain scheduling, published in Automatica en 2000, for insights on this issue. 

Thanks Edouard - that is exactly what I was looking for.

The answer  is no in general.  If all the controllers are designed for a single Lyapunov function, then the answer is yes.  

I have encountered a similar time-varying system with the following discrete dynamics:

x(k+1)=  (w1A1+w2A2+...wnAn) x(k),

where w1,...,wn are time-varying, non-negative scalars whose sum is 1, and A1...An are time-invariant state matrices.

A necessary and sufficient stability criterion for this system is called the "poly-quadratic stability condition". The stability test can be efficiently solved using a standard linear matrix inequality (LMI) solver.

Details of the stability test can be found in the following paper:

J. Daafouz and J. Bernussou, “Parameter dependent lyapunov func-tions for discrete time systems with time varying parametric uncertain-ties,” Systems & Control Letters, vol. 43, no. 5, pp. 355–359, 2001.

That paper looks great. Thanks for your help Wang.

It is great that the paper might help. I wish that your system can successfully pass the stability test. :-)

Best wishes,

Wang

If we kwon the bounds of the weighting coefficients, I think that even if the weights are time varying we can apply Horisberger Theorem, arriving to a feasibility problem in the vertex of the polytope. It is needed to assure that convexity is preserved in closed loop and that the parameter dependence is multi-afine. For further details see the book of Amato F., "Robust Control of Linear Systems Subject to Uncertain Time-Varying Parameters", Springer, 2006

Hi,

When stabilizing a nonlinear smooth system by interpolating local stable controllers (either transfer function or state feedback controllers) there are two conditions guaranteeing stability :

-Stability covering condition (every two neighbor controllers should have a non-empty intersection).

- The interpolating variable derivative should be small (condition on the variation of the interpolating variable)

Theses conditions are actually given, as mentioned by Edward, in papers by Rugh Shamma, Stilwell (not sure about the spelling) and Rugh

A.F.

If the polytope of controllers is LTI stable, it does not guarantee that the time-varying system will be stable.  One would need to put bounds on the way and the rate at which the variation occurs in order to satisfy the decrescency condition of the system.

If your controller guarantees point-wise stability (for each fixed parameter), you can transform it to a different controller that additionally guarantees stability in time varying case. Check the paper: 

Franco Blanchini, Daniele Casagrande, Stefano Miani, and Umberto Viaro. Stable

LPV realization of parametric transfer functions and its application to gain-scheduling

control design. Automatic Control, IEEE Transactions on, 55(10):2271–2281, 2010.

Why jump to multiple systems? Take just ONE single system first, which is stable for any fixed gain, say from 1 to 10.

Then, you let the gain vary at will within the “admissible domain.” It “might” remain stable, yet might also go unstable as well, depending of the specific gain variation rule.

This reminds us two famous conjectures of two great names in Control, Aizerman and Kalman, who thought otherwise, until people found counterexamples.

At some point in the past, I still thought I could “prove’ that the conjectures were right and I “almost’ proved it. Nevertheless, because of this failed attempt I learned so much ever since about nonstationary and/or nonlinear systems, passivity, stability, etc., that I call it “my best error ever.”

Bottom line, the general answer is no. For a specific nonstationary rule, you must find a way (Lyapunov-style methodology is the general way) to prove or disprove stability.

Time variance complicates the stability tests because the system stability has to be checked with passage of time. The original linear-combination of bounded-inputs is time-invariant; the time-varying coefficients can make either one or multiple bounded-inputs as a source of input stability.

Generally, the transition matrix of a time-varying system is tested for stability. First, I will test the system for homogeneity because the weights are time-varying linear-combinations. When any (time-varying) coefficient is changed by a certain factor - from 0 to 1 - the output must be scaled by that same factor in same time. This test implies that the system is homogeneous. Further, if the system is memoryless over any prescribed passage of time, the change in the mixing coefficient for a bounded input can give a bounded output over the same passage of time.

Is a time-varying linear-combination of multiple stable controllers in parallel also stable?

- The stability of time-invariant multiple controllers in parallel cannot convey the stability of that system with passage of time.