I understand the bandwidth limitations of feedback controllers in non-minimum phase systems (unstable zeros attract LHP poles to RHP). Classical controllers can deal with this rather easily by setting the bandwidth less than the RHP zero frequency etc. However when trying to implement state space control via an LQR, how do the penalising weights, R and Q, account for the bandwidth limitation? Is an LQR itself limited in non-minimum phase systems?
Thanks
A good question.
LQR control relies on the assumption that the state variables can be measured, that is, for a state space realization,(A,B,C,0), the output matrix is the identity matix, C=I.
Now the zeros of a MIMO system are the values of s such that the matrix
[ A-sI, B]
[C, D]
loses rank.For LQR, D=0 and C=I. Hence the matrix is
[ A-sI, B]
[I, 0]
which has full column rank for all s. Since the number of columns
Dear Daniel,
State feedback methods are different than what you refer to as classical control in the fact that, state feedback controllers allow you to locate closed-loop arbitrarily in the s-plane. In fact, you usually select closed-loop poles location, maybe using a design prototype, then you can calculate state feedback gains to achieve the targeted performance. Now, L-Q is a state feedback method, but it gives you the optimum gains in contrast to the former method. Q and R selection would result in different performance, but should render a stable system. The requirement for LQ method is that the plant should be detectable (observable), so that, you either measure all states or states are observable. So, I think NMP is not an issue in state feedback control methods including LQR.
I hope that helps,
A good question.
LQR control relies on the assumption that the state variables can be measured, that is, for a state space realization,(A,B,C,0), the output matrix is the identity matix, C=I.
Now the zeros of a MIMO system are the values of s such that the matrix
[ A-sI, B]
[C, D]
loses rank.For LQR, D=0 and C=I. Hence the matrix is
[ A-sI, B]
[I, 0]
which has full column rank for all s. Since the number of columns
Hi Daniel,
In first paper we fiwd, a linear-quadratic-regulator (LQR) is firstly design as a feedback controller and a feedforward controller is then proposed to eliminate the phase error emerge by the LQR controller during the tracking control. The feedforward controller is develop by implementing the zero phase error tracking control (ZPETC) technique which the main difficulty arises from the non-minimum phase system is have no stable inverse.
- PERFECT TRACKING CONTROL WITH DISCRETE-TIME LQR FOR A NON-MINIMUM PHASE ELECTROHYDRAULIC ACTUATOR SYSTEM
s2is.org/Issues/v4/n3/papers/abstract6.pdf
- CONTROLLER DESIGN FOR A NON-MINIMUM PHASE ELECTRO
dspace.jdvu.ac.in/bitstream/.../Acc.%20No.%20DC%201474.pdf
- IfA Fachpraktikum - Experiment 3.4 : Quad Tanks - ETH Zürich
https://www.ethz.ch/...systems-n.../script_quadruple_tank.pdf
Good Luck
Hi Daniel and All,
Sorry, but I see a little confusion here. If you can assume that you know the system and can use full-state feedback, there is no non-minimum-phase.
Minimum or non-minimum phase appear in input-output relations, where the particular state combination which gives you the particular output may result in "good" or "bad" phase.
One of the properties of full state-feedback that is used in LQR, besides giving the Optimal Solution, is high-gain stability. In other words, if you made a mistake and R is smaller than desired, the Gains could come out larger than needed or desired, yet this does not affect stability.
Moreover, if you increase the final gain even 100 or 1000 times, it does not affect stability, so the " bandwidth" in this case can be theoretically infinite. We say that LQR guarantees infinite gain-margin from above and 6 dB margin from below.
Disclaimer: I wrote "theoretically" because I would not suggest you use too high gains with any real-wold system, which may reach limitations and/or excite oscillations and nonlinearities that you would not even know they were there if not the high gain.
Nevertheless, the theoretical result is important.
Best regards,
Itzhak
Unstable systems apply Robust control theory and design a controller. LQR is preferable for stable systems.
Hi Daniel,
I fully agree with George V. I. Indeed the LQR controller remains essential for the stability system classes and other classes with an accompaniment of stabilization. In addition, the simplicity of design and well established know-how now playing in their side.
Best regards.
Any approach has its basic assumptions and LQR assumes availability of full-state.
As I wrote above, if you can assume this, then LQR not only guarantees stability, but also maintains it even if you made some terrible error and your gain is (or during operation changes to be) even 10 times what you intended.
Classical Control did not assume full-state availability and Robust Control goes back to assume that your Plant model, that you might have used for design, is only some nominal model. The real Plant might contain "unmodeled dynamics" that must be taken care of so that, at least, it does not affect stability and also performance can be maintained to some degree of satisfaction.
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