Hi
Consider the following closed loop generalized plant. P is the generalized plant, K is the controller to be designed and $\Delta$ matrix is the uncertainty matrix with the norm of smaller than one for all combination of the parameters.
I try to get the controller with H_inf design but the H_inf norm of the resulting closed loop from w_u to z_u is larger than 1. What does it mean?
I have tried to assume very power full actuation ( limitless). It never get smaller than 1. From theory means the controller is not robust. but why it is not possible to make it robust.
P.S: the real plant has two poles on origin.
I think this situation reveals a crucial deficiency of H infinity design -- it does not consider the loop phase at all, and the loop phase can be as important as loop gain (or norm, including H infinity norm). I think this deficiency is also common in other existing classical control designs,
The following well known example may illustrate the importance of designing both gain and phase of loop transfer function L(jω). This example is about the optimal L(jω) of quadratic optimal control, and this optimal L(jω) is proven to satisfy Kalman Inequality and thus must have values only outside of the unit circle that is centered at -1 point. Because large gain |L(jω)| is bad, the real optimal values of this L(jω) must be near the origin, and these values must have phase angles between 0⁰ to +-90⁰!
In my publications such as "Observer design -- A survey" of 2015 and "Robust Control Design" of 2004 (2nd Ed.), and in my answers to ResearchGate questions such as LQ control, I stated the following 4 points:
1. State feedback control (including partial state feedback control, its weakest case is static output feedback control if there are enough outputs) , can assign eigenvalues AND eigenvectors (can be partial) and can therefore improve feedback system performance AND robustness FAR BETTER than any other forms of control, including the control designed by H infinity, because eigenvectors can determine the sensitivity and robust properties of their corresponding eigenvalues. This is also why modern/state space control theory was dominant over classical control theory in the 1960s and 70s.
2. The problem with full state feedback control is that its loop transfer function Lkx(s) CANNOT be realized by the separately designed realizing observer. This existing design procedure is the well known separation principle, and this critical problem of LTR was raised by John Doyle, but the LTR solutions of the 1980s were not satisfactory, for example even the asymptotic LTR is invalid for non-minimum-phase systems or invalid for most systems. This should be the reason of the returned attention to H infinity design and other classical designs, starting in the 1980's.
3. It can be simply proved that the unsatisfactory LTR solutions of the 1980's were due to the following of separation principle. As a result, I developed a new design procedure NOT following separation principle and not always design a full state feedback control (in the case that Lkx(s) cannot be exactly realized). I will design instead a generalized state feedback control (GSFC) which unifies the existing full state feedback control and static output feedback control as its two extreme (strongest and weakest) cases. The design of this GSFC is fully based on (not separated from, as in separation principle), the observer parameters and parameter C of system (A,B,C).
4. By being able to freely choose the observer order, I can design an output feedback compensator (OFC) that is valid for almost all systems (including most of the non-minimum-phase systems), and that can generate the GSFC signal. The actual OFC order, which equals the number of the estimated states/additional outputs, is chosen based on the actual system conditions and actual design requirements, unlike the fixed maximum order for full state feedback control and the fixed zero order for static output feedback control. The critical advantage of this OFC is that Lkx(s) of its GSFC is guaranteed realized exactly. Thus a satisfactory (general and exact) LTR solution is finally achieved, and the need of H infinity design of the 1980s is finally obviated.
Hi Chia-Chi Tsui
Thanks you for your answer. Can you refer me to some publication that I can read more about it?
I have read your answer multiple times and it is pretty difficult to me to understand.
"Observer Design -- A Survey", Int. J. of Automation and Computing, vol. 12, no. 1, pp. 50-61, 2015.
Robust Control System Design – Advanced State Space Techniques. First Edition: Marcel Dekker Inc., NY, 1996; Second Edition: Taylor Francis 2004.
Let me explain why I (and many others) stated in the second paragraph of my precious answer, that " large loop gain |L(jω)| is bad":
If only the magnitude of L(s) (and of overall transfer function) is considered, then large loop gain generally means low overall gain and therefore low performance.
Large loop gain also implies high likelihood of instability, because the feedback system poles will approach all zeros (including unstable zeros) at high loop gain.
Large loop gain also requires high physical power for actuation, and therefore higher chance of saturation and component break downs.
I suspect that these are the reasons that made the above-unity sensitivity in your project, because if only the magnitude of L(s) (and of sensitivity function) is considered, then low loop gain means high sensitivity function value.
In other words, if only the gains (but not the phase) is considered, then it will be almost impossible to accommodate both high performance and low sensitivity (or high robustness), let along adjust effectively the trade off between these two critical properties.
However, this accommodation and adjustment is indeed possible using state space techniques such as the optimal control example mentioned in the previous answer, because the phase angles can be cleverly placed between 0⁰ to +-90⁰ at low loop gain frequency. This is again a proof of the superiority of state feedback control.
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