1 Robustness Analysis

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Robustness Analysis The reason we are studying phase portraits of linear systems is because we plan to linearize the true nonlinear system of interest and then extrapolate the results obtained for the
Robustness Analysis The reason we are studying phase portraits of linear systems is because we plan to linearize the true nonlinear system of interest and then extrapolate the results obtained for the linear system to the parent nonlinear system. It follows that it is important to know to which extent the results obtained for linear systems are robust, in the sense that the qualitative statements made for them remain true after perturbations. Eigenvalues are continuous functions of the matrix entries when they have multiplicity one. It follows that nodes, focus and saddle points are structurally stable with respect to small perturbations. Imag. Real. I Figure. Robustness of Nodes, Focus and Saddle Points This is not the case for centers, where small perturbation can make them to have positive or negative real parts. In certain cases, complex eigenvalues (always in pairs, because complex conjugates) can become multiple after small perturbation and then produce a catastrophic change of the phase portrait. Imag. Real. Figure. Lack of Robustness of Circles If eigenvalues are zero then changes can be even more dramatic as we will see in the lecture on bifurcations. Which tools are available to analyze robustness? . Stability Radius In this case one tries to answer the question about at which size, there are perturbations that can destabilize our nominal design. One answer is to look at the stability radius. One has in mind the following illustration. Definition. Given the matrix triple The stability radius is the number nxn nxp qxn ( A B, C) R R R,. r k = such that the spectrum of matrices of the form r ( A, B, C) k A + B Δ C is in the left hand side of the complex plane for any whose norm is less than r k. Δ { C} K pxq, K R, The main reason the complex stability radius (i.e Delta is allowed to be complex) is useful is because it is an easy to calculate robustness measure. It might produce conservative estimated though. On the other hand, the complex case has special implications for the analysis of nonlinear systems. The following theorem holds for the calculation of the complex stability radius. r C = / { maxw R, C( wi A) B } This result is closely related to the so called small gain theorem.. Spectral Value Sets In the previous case, we looked at the robustness of the stability property. In other cases one is interested on whether the nominal design will keep the eigenvalues in a desired region under the action of perturbations of given size. One answer is the so called spectral value sets. Definition: Spectral Value Set σ ( A, B, C, ρ) is a set of the complex plane given by the union of the spectra of all matrices of the form A + BΔC with Δ such that its norm is smaller than ρ. It follows that if σ ( A, B, C, ρ) remains in the desired region for appropriate values of ρ, then the design can be seen as robust enough for the application. Complex Spectral Value Sets can be easily computed by looking at the contour plot of the norm of the transfer function defined by the triple ( A, B, C). Indeed, it holds. { A ρ} B (, B, C, ρ) = s I, C( si ) σ A /. In other words, one must plot the surface C( si A) s, and then look at the contour plots given by eigenvalues of the matrix A. Example.Two Mass Spring System Let us look at the following system B as function of the complex number / ρ. Note that the set includes the In some rough approximation of the spring behavior, the system can be described by the equations. This gives us the matrices (with k=m=m) A=[ ; ;- ; - ]; B=[;;;]; C=[ ]; D=; The eigenvalues of A are +/-.44i and (twice), ie on the imaginary axis. Thus we do not expect a robust open loop system. For instance, see the behaviour for x=[-,,,] that generates a circle and x=[-,,,] that produces resonance. dx x dx x dx x dx x Figure 3. Open Loop Trajectory for x=[-,,,] and x=[-,,,]. However, if we apply a static feedback law u=kx, designed using LQR, we see that the system becomes much more stable because the poles moves into the left hand side of the plane. Open and Closed Loop Poles.5.5 Imag(s) Real(s) Figure 4. Open Loop Poles (circles) and Closed Loop Poles (x) for different values of control cost in the LQR design. The lower the cost, the further away the poles moves from the imaginary axis. The stable behavior can be seen in the picture below. 4 dx x dx x Figure 5. Closed Loop Trajectory with LQR controller But how robust are these pictures? This can be assessed using the stability radius calculation and the spectral value sets. Let us look at the disturbance w(t) depicted in the Figure. This suggests that we should analyze the effect of the structured disturbance given by the matrices Bd=[;;;]; Cd=eye(4); on the closed loop system (A-BK). Calculations show that the complex stability radius rc(a-bk,bd,cd)=.57 and that the size of w(t) must be at least /57 before the system becomes unstable due to this disturbance. The spectral value sets also show a relatively robust system. Note the value for which the contours cross the imaginary axis is equal to /rc Contours Figure 6. Spectral value sets of the closed loop system with respect to disturbance w(t)
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