# 6.241 Dynamic Systems and Control

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1 6.241 Dynamic Systems and Control Lecture 17: Robust Stability Readings: DDV, Chapters 19, 20 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology April 6, 2011 E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

2 Motivation All analytical methods for control design are based on a model of the system to be controlled. Such a model does not necessarily represent a perfect description of the system, for several reasons, for example: The complexity of a real physical system can not be handled well by mathematical models. Even in the case in which a perfect model can be designed for a given system, in general the same model is not a perfect description under all operating conditions. The experimental identification of a system s model (including validation of a mathematical model) is very difficult for open-loop unstable plants, for obvious reasons. Even for stable plants, an accurate experimental measurement of high-frequency behavior is very hard to get. Certain characteristics of the model may not be amenable to easy control design (e.g., nonlinearities, or very high order or fast dynamics). In these cases, it may be preferable to formulate a simple linear model of the system, that can be used as the basis for control design. E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

3 System uncertainty and uncertainty models P Uncertainty model "real" system P nominal system System uncertainty Still, even though a perfect model of the system is not available, it is desired to design an automatic control system that performs according to some specifications not only for the given model, but also for the real system. In order to take such uncertainty into account, we will first come up with an uncertainty model, consisting of 1 2 A nominal model; A set of models that is guaranteed to contain the system uncertainty, and is easier to handle. and then design a control system that meets the stability and performance specifications not only for P, but also for all other possible models in the uncertainty model. E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

4 Multiplicative uncertainty Multiplicative uncertainty models are of the form P (s) = (1 + W 2 (s)δ(s))p(s), in which the frequency weight W 2 is a given stable transfer function, and Δ is an unknown stable transfer function, such that Δ < 1. Note that since ( ) 1 P (s) W 2 (s) P(s) 1 = Δ(s), and Δ 1, it must also be true that ( ) 1 P (jω) < 1 W P(jω) 1 2 (jω) for all frequencies ω, i.e., that P (jω) P(jω) 1 < W 2(jω), ω R. So the multiplicative uncertainty model is a description of how much the ratio of the real and nominal transfer function is away from being equal to E. one. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

5 Multiplicative uncertainty Among other things, multiplicative uncertainty is useful when the gain of P is uncertain. For example, consider the uncertain system P (s) = γg (s), γ [γ, γ + ], where G (s) is a known transfer function. We can represent the same set of systems using a multiplicative uncertainty model with and γ + γ + P(s) = γ 0 G (s), γ 0 =, 2 W 2 (s) = γ + γ. γ + + γ E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

6 Additive uncertainty Additive uncertainty models are of the form P (s) = P(s) + W 2 (s)δ(s). Additive uncertainty is useful when the numerator of P contains uncertain terms, such as zero locations. For example, consider the case in which P (s) = (s m βs l )G(s), β [β, β + ]. We can represent the same set of systems with an additive uncertainty model of the form P(s) = (s m β 0 s l )G (s), β 0 = β + β +, 2 and W 2 (s) = β + β s l. β + + β E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

7 Feedback Uncertainty models Feedback uncertainty models are of the form P(s) P (s) =. 1 + W 2 (s)δ(s)p(s) Feedback uncertainty is useful when the denominator of P contains uncertain terms, such as damping coefficients. For example, consider the case in which G (s) P (s) =, α [α, α + ]. s n αs l We can represent the same set of systems with a feedback uncertainty model of the form and G (s) α + α + P(s) =, α 0 =, s n α 0 s l W 2 (s) = α + α s l. α + + α E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

8 Nyquist stability criterion Let Γ be a closed path in C, which does not go through any of the zeroes/poles of a function F : C C. The image of Γ under F will encircle the origin N = Z P times, where Z and P are, respectively, the numbers of zeroes and poles of F inside the contour Γ. Consider F (s) = 1 + L(s). The poles of F are the poles of the loop transfer function L; the zeroes of F are the closed-loop poles (i.e., poles of (1 + L) 1 ). If Γ is the D -contour, traversed clockwise, then the F Γ contour will encircle the origin clockwise N = Z P times. In other words, the closed loop will be stable if and only if P = Z N, i.e., if the number of times L Γ encircles the -1 point is exactly equal to the number of open-loop unstable poles. E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

9 Robust stability In the case of additive uncertainty, L = (P 0 + W Δ)K = L 0 + W ΔK, the Nyquist plot will not encircle the 1 point if W (jω)k(jω) < 1 + L 0 (jω), i.e., if W (jω)k(jω) < 1, ω R. 1 + L 0 (jω) Another way to look at this is by this equivalence: W (jω)k(jω) min 1 + Δ(jω) > 0, Δ(jω) P 0 (jω)k(jω) W (jω)k(jω) 1 + L 0 (jω) < 1, ω R. ω R E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

10 Proof 2 1: W (jω)k(jω) W (jω)k(jω) 1 + Δ(jω) 1 Δ(jω) 1 + P 0 (jω)k(jω) 1 + P 0 (jω)k(jω) W (jω)k(jω) P0 (jω)k(jω) 1 2: Assume there exists ω 0 for which the ratio is 1. Define W (jω 0 )K jω0 = ae jφ, 1 + P 0 (jω 0 )K(jω 0 ) if Δ(jω 0 ) = 1 a e jφ jπ, the system is unstable. This can be achieved by choosing Δ(s) = ± 1 s α, a s + α since one can always find α such that ± jω0 α = e jφ jπ. jω 0+α E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

11 Linear Fractional Description Rewrite the block diagram in the following way, i.e., isolating Δ [standard G Δ block diagram] The transfer function of the non-δ part of the block diagram can be written as [ ] [ ] [ ] u G uy G uw y =, z G zy G zw w where u and y are, respectively, the input and output to Δ. The system is assumed to be nominally internally stable (a stabilizing controller may be embedded in G ), hence all blocks in G are stable. If the interconnection is closed as a feedback on Δ, how do the stability properties of the system change? Given y = Δu, we can solve for the closed-loop transfer function w z: Δu = y = ΔG uy y + ΔG uw w y = (I ΔG uy ) 1 ΔG uw w and hence ( ) z = G zy y + G zw w = (I ΔG uy ) 1 ΔG uw + G zw w. E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

12 Unstructured Small-Gain Theorem Let M = G uy be the transfer function describing G as seen by Δ. The robust stability problem is hence equivalent to establishing the stability of for any stable Δ with Δ H < 1. (I ΔM) 1 Δ = Δ(I MΔ) 1, Theorem ( Unstructured Small-Gain Theorem) Let Γ = {Δ stable : Δ H < 1. If M is stable, then (I MΔ) 1 and Δ(I MΔ) 1 are stable for all Δ Γ if and only if M H < 1. E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

13 Proof sufficiency To prove sufficiency, it is enough to show that if M H < 1, (I MΔ) has no zeros in the closed RHP. For any x = 0, and s + in the closed RHP, [I M(s + )Δ(s + )]x 2 x 2 M(s + )Δ(s + )x 2 x 2 σ max [M(s + )Δ(s + )] x 2 x 2 M H Δ H x 2 > 0 E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

14 Proof necessity We want to prove that if σ max (M(jω 0 )) > 1 for some ω 0, then one can construct a Δ with Δ H < 1 that makes the closed-loop system unstable. Consider the singular-value decomposition M(jω 0 ) = UΣV, with σ 1 > 1. Then one can choose Δ(jω 0 ) = V 1/σ 1 0 U.... Clearly, I M(jω 0 )Δ(jω 0 ) = I UΣV V 1/σ 1 0 U... 1/σ 1 0 I 0 U = U 1 = U U, i.e., a singular matrix. It remains to find a transfer function Δ such that Δ(jω 0 ) has the desired form. E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

15 Performance as Stability Robustness A performance criterion, e.g., disturbance rejection, can be stated as a bound on the H norm of a certain transfer function. This is formally the same as the robust stability problem. (i.e., take the certain transfer function as M) E. Frazzoli (MIT) Lecture 17: Robust Stability April 6, / 15

16 MIT OpenCourseWare J / J Dynamic Systems and Control Spring 2011 For information about citing these materials or our Terms of Use, visit:

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