Dear Joseph Jose,

I suggest you to see links and attached files on topic.

https://www.researchgate.net/publication/258985778_Survey_of_gain-scheduling_analysis_and_design

https://www.adrianilka.eu/img/PhDthesis_Ilka.pdf

https://www.hindawi.com/journals/mpe/2014/906367/

https://users.isy.liu.se/rt/johanl/2012_TAC_LPVMPC.pdf

Best regards

Dear Harald Kirchsteiger

thank you for your valuable response

can global stability be assured if i ensure these two

1. local stability of all the controllers .

2. avoid frequent switching between controller blocks by adding a hysteresis block which force the controller to stay in a particular block for a minimum time period.

Hi Dr. Joseph Jose

• Could you provide a bit more info about the behavior of the original system?
• Does the system behave highly nonlinear or slightly nonlinear in the desired operating range, or a mixed of both throughout the operating range?
• How accurate can you approximate the original system with 7 linear models?
• Are the 7 linear models obtained from 7 equally-spaced operating points?

Hi Dr. Yew-Chung Chak

thank you for the response

1. System is a mix of highly nonlinear and slightly non linear throughout the operating range

2. linear models can approximate the original system with less than 20% mean squared error

3. No linear models are not obtained from 7 equally spaced operating points

the system response plot is shown in figure there is a dead band in the middle from -0.15 to 0.15

the seven linear systems are obtained as follows based on command signal

linear system 1----(command signal -1 to -0.5)

linear system 2----(command signal -0.5 to -0.25)

linear system 3----(command signal -0.25 to -0.15)

linear system 4----(command signal -0.15 to 0.15)

linear system 5----(command signal 0.15 to 0.25)

linear system 6----(command signal 0.25 to 0.5)

linear system 7----(command signal 0.5 to 1)

the slight non-linearities across theta is ignored

For an extensive stability analysis of the system, I suggest to study the theory of "switching system" stability . This theory cover your problem, but the stability is defined in sense of Filipov, etc.

Hi Dr. Joseph Jose

Since there are 7 linear models, if you can show that there exists a common positive definite matrix P such that for all i = 1, 2, ..., 7 and j = 1, 2, ..., 7,

(Ai – Bi·Kj)]T·P·(Ai – Bi·Kj) – P < 0,

then the equilibrium of the discrete-time MPC system maybe globally asymptotically stable.

Dear Dr.Yew-Chung Chak ,

But state-space models have different Dimensions (linear systems 1,2,3 and 5,6,7 have 9 states and linear system 4 has only 8 states)

will I be able to ensure global stability if the following two conditions are met?

1. local stability of all the controllers .

2. (switching stability) avoid frequent switching between controller blocks by adding a hysteresis block which forces the controller to stay in a particular block for a minimum time period.

Hi Dr. Joseph Jose

What caused linear system #4 to have 8 states only? The dead band? Can you augment linear system #4 by creating another state?

Stability conditions are derived from the math. What is the mathematical representation of the local stability of all controllers?

It is not recommended to have abrupt switching between the controllers. That's why you have the gain schedule. Can you make the switching between controllers to occur in smooth transitions?

Hi Dr.Yew-Chung Chak

due to zero input signal and dead band, a state representing a particular valve opening is zero in system 4.

Local stability of controllers were analyzed in MATLAB linear system analysis.

Switching between controllers can be made smooth by making them stay in a particular mode for a finite time period (Dwell time) by compromising the controller performance.

Hi Dr. Joseph Jose

Since a particular valve opening is zero in system 4, you can probably augment the state matrix and input matrix by having zero rows. But the augmented system maybe interpreted as non-controllable. Can you try placing a very small positive coefficient (0+) for the augmented input state?

As mentioned by other RG members, local stability cannot serve as proof for global stability unless you can show this:

(Ai – Bi·Kj)]T·P·(Ai – Bi·Kj) – P < 0

I would rather suggest to have gain scheduling for the entire prediction horizon. For example, if you have initial time t0 and final time be tf, then better have N samples over this entire prediction horizon for some step size h. In this case, you 'll be having N linear models computed over entire prediction horizon and you won't have any sort of a "long" dead-band (though there may practically be some but almost is negligible in most of the cases if h is chosen suitably). Moreover, better check whether for each model, the solution of the ARE turns out to be feasible with P being positive definite or not. Better incorporate built-in integrator in MPC model structure to mimic prediction error.

Generally, the global stability has to be inferred from extensive simulations and mainly depends on how the controller is scheduled i.e., use interpolation, linearization scheduling or other methods, how many and which operating points to use as a basis for the scheduling, etc.,

While considering the classical gain scheduling methods, the key issues are i) the fact that transitions among operating conditions are not addressed in the design, and ii) shortcomings of the linearization’s. Hence, extensive offline testing is required to establish global stability and performance guarantees. An often-used rule-of-thumb states that the scheduling variable should only vary slowly to prohibit the introduction of additional dynamics.

Generally, the scheduling is based intuitively on physical variables in the plant. In case of the quasi-LPV, a coordinate transformation has to be applied. As this approach is conservative, this might lead to infeasible controller design problem, i.e. no feasible controller may be available. Quasi-LPV techniques on the other hand, enable global stability as well as performance and robustness analysis.