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- Politecnico di Milano
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- Aerospace Engineering
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- AEROSPACE CONTROL SYSTEM
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- English
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- Notes · Complete set
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NotesComplete setEnglish
Completed notes of the course
Complete course materials for AEROSPACE CONTROL SYSTEM in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Aerospace Control Systems Lecture notes (A.Y. 2023/24) Typeset from handwritten notes (pp. 2–128) Figures are cropped from the original handwritten notes. Boxes marked Correction flag errors found in the original. Grey Ref. lines point to the books listed in the Bibliography.
AEROSPACE CONTROL SYSTEMComplete set
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Complete course materials for AEROSPACE CONTROL SYSTEM in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Aerospace Control Systems Lecture notes (A.Y. 2023/24) Typeset from handwritten notes (pp. 2–128) Figures are cropped from the original handwritten notes. Boxes marked Correction flag errors found in the original. Grey Ref. lines point to the books listed in the Bibliography.
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Aerospace Control Systems Lecture notes (A.Y. 2023/24) Typeset from handwritten notes (pp. 2–128) Figures are cropped from the original handwritten notes. Boxes marked Correction flag errors found in the original. Grey Ref. lines point to the books listed in the Bibliography. Contents 1 Course introduction 5 2 Linear time-invariant systems 5 2.1 State-space representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Scalar case (n = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 General case (n > 1): Lagrange formula . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Computing e At . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Equivalence and coordinate transformations . . . . . . . . . . . . . . . . . . . . . . 7 2.6 Superposition principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 From state space to transfer functions 8 3.1 Poles and zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Response to a given input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Stability 9 4.1 Equilibria and stability definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 Lyapunov’s indirect method (linearization) . . . . . . . . . . . . . . . . . . . . . . . 10 4.3 Lyapunov’s direct method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.4 Pendulum revisited with Lyapunov’s direct method . . . . . . . . . . . . . . . . . . 11 4.5 Region of attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.6 LaSalle’s invariance…
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5.4 Control sensitivity Ref. [SP] §2.7.3 (p. 63): KS is the control sensitivity The behaviour of u(t) matters, especially during transients, because of saturation risk (e.g. an aileron cannot be deflected beyond its stroke end, or faster than a certain rate). Define R(s) Q(s) = , (15) 1 + L(s) the control sensitivity function, which represents the effect of the set point on the control input, of the disturbance on the control input and, up to the sign, of the measurement error on the control input. It holds 1 ω ≪ ωc | R( jω )| | G ( jω )| | Q( jω )| = ≃ |1 + L( jω )| | R( jω )| ω ≫ ωc Up to the crossover, the control sensitivity is inversely proportional to the frequency response of the plant; above the crossover it coincides with the frequency response of the controller. s+1 10 10 Example 5.2. PI controller R(s) = , plant G (s) = 2 , so L(s) = R(s) G (s) = . s ( s + 1) s ( s + 1) Verify at home that the feedback loop is AS (graph in the slides). 6 Norms of signals and systems Ref. [K] §5.1 (p. 195); [SP] §A.5 (p. 550), §4.10 (p. 162) Reference: slides “Performance 2”. Vector norms. For u(t) = [u1 (t), . . . , un (t)]T : ∥u(t)∥∞ = max |ui (t)|, i =1,...,n q q ∥u(t)∥2 = u ( t )T u ( t ) = u21 (t) + · · · + u2n (t) (Euclidean or 2-norm). Signal norms. ∞ rZ ∥u∥L∞ = sup ∥u(t)∥, ∥ u ∥ L2 = uT (t)u(t) dt. t ≥0 0 Sometimes a truncated version of the signal is used: ( u(t) 0≤t≤τ L e = { u : u τ ∈ L, ∀ τ ≥ 0}, uτ (t) = 0 t>τ (the extended space). 19
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Note that, while for the gain a change in δ shifts the Bode plot up and down in a uniform way, for pole uncertainty the response varies with frequency. Plotting |W ( jω )| with rτ = 1 (just for simplicity): +20 dB/dec at low frequency, then flat at 0 dB beyond the nominal frequency of the pole 1/|τ̄ |. Bode magnitude of W (s) for pole uncertainty. 10.5 Zero uncertainty Ref. [SP] §7.7.3 (p. 302) GP (s) = (1 + τP s) G0 (s), τm ≤ τP ≤ τM . Keeping all the conventions set for pole uncertainty, after simple passages: sτ̄rτ GP (s) = Ḡ (s) 1 + W (s)δ , Ḡ (s) = (1 + sτ̄ ) G0 (s), W (s) = . (39) τ̄s + 1 Frequency response: at low frequencies the system is basically insensitive; at high frequencies it is sensitive to changes of δ (the threshold is the natural frequency of the nominal zero). 10.6 Multiple uncertain parameters Ref. [SP] §7.4.1 (p. 275) How do we deal with a transfer function with multiple uncertain parameters and study their simultaneous effect on the dynamics? Simple example: k 1 GP ( s ) = · , τs + 1 τ2 s + 1 with τ2 known and k, τ unknown, in known intervals k m ≤ k ≤ k M , τm ≤ τ ≤ τM . We can define δk (−1 ≤ δk ≤ 1) and δτ (−1 ≤ δτ ≤ 1) and replace k, τ with representations similar to the ones seen previously. k̄ Nominal model Ḡ (s) = with the gain block rk δk and the pole feedback block W (s)δτ . (τ̄s + 1)(τ2 s + 1) What about the frequency response of the system? Effect of k and τ on the Bode magnitude: first pole 1/|τ |, second pole 1/|τ2 |. 37
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RHP poles. Lecture of 10/05/2024. For RHP zeros, ωc < z(1 − 1/M ), with M the high-frequency gain: this is a hard limit, the crossover cannot (in any way) be greater than the zero z. What about poles in the RHP? Let WF (s) be a stable weighting function; for closed-loop stability, since F ( p) = 1, max |WF ( jω ) F ( jω )| ≥ |WF ( p)|, ω and this quantity is involved in robust stability conditions (recall RS: < 1), so |WF ( p)| < 1. Note: these two conditions (RHP zero and RHP pole) may be conflicting. With s 1 1 M WF (s) = + , = M , ωc M WF (s) ω s+1c p 1 M |WF ( p)| < 1 ⇒ + < 1 ⇒ ωc > p ωc M M−1 An exponentially unstable system with both p and z: the admissible range for ωc lies between them. The situation is manageable, but challenging if the range is narrow. Balancing a pen on the top of your finger falls under this last example (a short stick means that p and z are close: the problem is more challenging). 16 MIMO LTI systems Ref. [SP] Ch. 3 (p. 69), §4.1 (p. 123) We will spend some time dealing with definitions. 16.1 State-space models With p outputs and m inputs: y1 u1 x1 ( .. .. ẋ = Ax + Bu y = . ( p × 1), u = . ( m × 1), x = ... (n × 1), y = Cx + Du yp um xn with A (n × n), B (n × m), C ( p × n), D ( p × m). For input i and output j, let bi be the i-th column of B, c j the j-th row of C and d ji the ( j, i ) element of D: ( ẋ = Ax + bi ui y j = c j x + d ji ui is now a regular SISO system. 55
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[K] H. K. Khalil, Nonlinear Systems, 3rd ed., Pearson New International Edition, 2014 (page numbering as in the 3rd edition, 2002). [ÅM] K. J. Åström, R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers, Princeton University Press, 2008 (electronic version v2.10b). [SLJ] B. L. Stevens, F. L. Lewis, E. N. Johnson, Aircraft Control and Simulation, 3rd ed., Wiley, 2016. 74
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