Closed-Loop System Analysis

Topic #24 16.30/31 Feedback Control Systems Closed-loop system analysis • Bounded Gain Theorem • Robust Stability 16.30/31 24–1 SISO Performance Objectives • Basic setup: di r e Gc(s) u do G(s) y − n Where the tracking error can be written as: e = r − (y + n) = r − (do + G(di + Gce) + n) = S(r − do − n) − SGdi y = T (r − n) + Sdo + SGdi with S = (I + L)−1 L = GGc, T = L(I + L)−1 • For good tracking performance of r (typically low frequency), require e small ⇒ IS(jω)I small ∀ 0 ≤ ω ≤ ω1 • To reduce impact of sensor noise n (typically high frequency), require ⇒ IT (jω)I small ∀ ω ≥ ω2 • Since T (s) + S(s) = I ∀ s cannot make both IS(jω)I and IT (jω)I small at the same frequen­ cies ⇒ fundamental design constraint 16.30/31 24–2 SISO Design Approaches • There are two basic approaches to design: • Indirect that works on L itself – classical control does this • Direct that works on S and T • For the Indirect, note that • If |L(jω)| » 1 ⇒ S = (1 + L)−1 ≈ L−1 ⇒ |S| « 1, |T | ≈ 1 • If |L(jω)| « 1 ⇒ S = (1 + L)−1 ≈ 1 and T ≈ L ⇒ |T | « 1 So we can convert the performance requirements on S, T into spec­ ifications on L (A) High loop gain � Good command following & dist rejection (B) Low loop gain � Attenuation of sensor noise. • Of course, we must be careful when |L(jω)| ≈ 1, require that arg L = ±180◦ to maintain stability 16.30/31 24–3 • Direct approach works with S and T . Since e = r − y, then for perfect tracking, we need e ≈ 0 ⇒ want S ≈ 0 since e = Sr + . . . • Sufficient to discuss the magnitude of S because the only require­ ment is that it be small. • Direct approach is to develop an upper bound for |S| and then test if |S| is below this bound. |S(jω)| < 1 |Ws(jω)| ∀ω? or equivalently, whether |Ws(jω)S(jω)| < 1, ∀ω • Typically pick simple forms for weighting functions (first or second order), and then cascade them as necessary. Basic one: Ws(s) = s/M + ωB s + ωB A Fig. 1: Example of a standard performance weighting filter. Typically have A « 1, M > 1, and |1/Ws | ≈ 1 at ωB 16.30/31 24–4 • Example: Simple system with Gc = 1 150 G(s) = (10s + 1)(0.05s + 1)2 • Require ωB ≈ 5, a slope of 1, low frequency value less than A = 0.01 and a high frequency peak less than M = 5. Ws = s/M + ωB s + ωB A Fig. 2: Want |SWp | < 1, so we just fail the test • Graph testing is OK, but what we need is a an analytical way of determining whether |Ws(jω)S(jω)| < 1, ∀ω • Avoids a graphical/plotting test, which might be OK for analysis (a bit cumbersome), but very hard to use for synthesis 16.30/31 24–5 Bounded Gain • There exist very easy ways of testing (analytically) whether |S(jω)| < γ, ∀ω • Critically important test for robustness • SISO Bounded Gain Theorem: Gain of generic stable system1 ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) is bounded in the sense that Gmax = sup |G(jω)| = sup |C(jωI − A)−1B + D| < γ ω ω if and only if (iff) 1. |D| < γ 2. The Hamiltonian matrix H= A + B(γ 2 I − DT D)−1 DT C − C T (I + D(γ 2 I − DT D)−1 DT )C −AT − C T D(γ 2 I − DT D)−1 B T has no eigenvalues on the imaginary axis. 1 MIMO B(γ 2 I − DT D)−1 B T result very similar, but need a different norm on the TFM of G(s) 16.30/31 24–6 • Note that with D = 0, the Hamiltonian matrix is # " 1 T A BB γ2 H= −C T C −AT • Eigenvalues of this matrix are symmetric about the real and imag­ inary axis (related to the SRL) • So supω |G(jω)| < γ iff H has no eigenvalues on the jω-axis. • An equivalent test is if there exists a X ≥ 0 such that 1 AT X + XA + C T C + 2 XBB T X = 0 γ and A + γ12 BB T X is stable. • This is another Algebraic Riccati Equation (ARE) • But there are some key differences from the LQR/LQE ARE’s 16.30/31 24–7 Typical Application • Direct approach provides an upper bound for |S|, so must test if |S| is below this bound. 1 |S(jω)| < ∀ω? |Ws(jω)| or equivalently, whether |Ws(jω)S(jω)| < 1, ∀ω • Pick simple forms for weighting functions (first or second order), and then cascade them as necessary. Basic one: Ws(s) = s/M + ωB s + ωB A • Thus we can test whether |Ws(jω)S(jω)| < 1, ∀ω by: 1. Forming a state space model of the combined system Ws(s)S(s) 2. Use the bounded gain theorem with γ = 1 3. Typically use a bisection section of γ to find |Ws(jω)S(jω)|max • For our simple example system 150 G(s) = (10s + 1)(0.05s + 1)2 Gc = 1 • Require ωB ≈ 5, a slope of 1, low frequency value less than A = 0.01 and a high frequency peak less than M = 5. • In this case γ = 1.02, so we just fail the test - consistent with graphical test. 16.30/31 24–8 Issues • Note that it is actually not easy to find Gmax directly using the state space techniques • It is easy to check if Gmax < γ • So we just keep changing γ to find the smallest value for which we can show that Gmax < γ (called γmin) ⇒ Bisection search algorithm. • Bisection search algorithm (see web) 1. Select γu, γl so that γl ≤ Gmax ≤ γu 2. Test (γu − γl )/γl < TOL. Yes ⇒ Stop (Gmax ≈ 12 (γu + γl )) No ⇒ go to step 3. 3. With γ = 12 (γl + γu), test if Gmax < γ using λi(H) 4. If λi(H) ∈ jR, then set γl = γ (test value too low), otherwise set γu = γ and go to step 2. • This is the basis of H∞ control theory. 16.30/31 24–9 Appendix: Sketch of Proof • Sufficiency: consider γ = 1, and assume D = 0 for simplicity • Now analyze properties of this special SISO closed-loop system. r u G(s) y y2 GT (−s) + � G(s) := A B C 0 � and G˜(s) ≡ GT (−s) := � −AT −C T BT 0 � • Note that u/r = S(s) = [1 − G˜G]−1 • Now find the state space representation of S(s) ẋ1 = Ax1 + B(r + y2) = Ax1 + BB T x2 + Br ẋ2 = −AT x2 − C T y = −AT x2 − C T Cx u = r + B T x2 � ⇒ ẋ1 ẋ2 � � T �� A BB x1 T T −C C −A x2 � � � � x1 u = 0 BT +r x2 = � � + B 0 � r ⇒ poles of S(s) are contained in the eigenvalues of the matrix H. 16.30/31 24–10 • Now assume that H has no eigenvalues on the jω-axis, ⇒ S = [I − G˜G]−1 has no poles there ⇒ I − G˜G has no zeros there • So I − G ˜ G has no zeros on the jω-axis, and we also know that I − G*G → I > 0 as ω → ∞ (since D = 0). • Together, these imply that I − G*G = I − GT (−jω)G(jω) > 0 ∀ ω • For a SISO system, condition (I − G*G > 0) is equivalent to |G(jω)| < 1 ∀ ω which is true iff Gmax = max |G(jω)| < 1 ω • Can use state-space tools to test if a generic system has a gain less that 1, and can easily re-do this analysis to include bound γ.