6.241 Dynamic Systems and Control Lecture 12: I/O Stability Readings: DDV, Chapters 15, 16 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology March 14, 2011 E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 1 / 12� � Introduction Last week, we looked at notions of stability for state-space systems, with no inputs. Now we want to consider notions of stability under the effect of a (forcing) input. Central to the discussion is the notion of norm of a signal—which is just the same we already discussed, when considering signals as infinite-dimensional vectors. In the following, let w : T Rn, with w(t) = w1(t) w2(t) . . . wn(t).→ E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 2 / 12Signal norms ∞-norm: Peak magnitude �w�∞ = sup t∈T �w(t)�∞ = sup t∈T max i=1...n |wi (t)| 2-norm: (Square root of the) Energy �w�2 2 = ⎧ ⎪⎨ ⎪⎩ � k∈Z w[k]�w[k] = � k∈Z �w[k]�2 2 (DT) �∞ −∞ w (t)�w(t) dt = �∞ −∞ �w(t)�2 2 dt (CT) Power (NOT a norm!) Pw = ρ2 w = ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ lim N→+∞ 1 2N N� k=−N w [k]�w [k] = lim N→+∞ 1 2N N� k=−N �w [k]�2 2 (DT) lim T →+∞ 1 2T � T −T w(t)�w(t) dt = lim T →+∞ 1 2T � T −T �w (t)�2 2 dt (CT) 1-norm: Action �w�1 = ⎧ ⎪⎨ ⎪⎩ � k∈Z �w [k]�1 (DT) �∞ −∞ �w(t)�1 dt (CT) E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 3 / 12Some examples Let w(t) = w¯ , ∀t ∈ T. Then, �w�∞ = |w¯ |; �w�2 = +∞; ρw = |w¯ |; �w�1 = +∞. we−atLet w(t) = ¯ , ∀t ∈ R≥0, and a > 0. Then, �w�∞ = |w¯ |; �w�2 = |w¯ |/√2a; ρw = 0; �w�1 = |w¯ |/a. E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 4 / 12System Norms Recall that a I/O model of a system is an operator mapping an input signal u to an output signal y, i.e., y = Su. We can define an induced norm for a system in exactly the same way as we did for matrices, i.e., �S�p,ind := sup �Su�p u=0��u�p We will see how to compute system norms later. E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 5 / 12Input-Output stability Definition (Input-Output stability) A system with I/O model S is p-stable (or �p-stable, or Lp -stable), if and only if its p-induced norm is finite, i.e., �S�p,ind < ∞. In particular, a system is Bounded-Input, Bounded-Output stable if and only if it is ∞-stable. Example: an integrator is not BIBO stable, and not p-stable for any p. E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 6 / 12� BIBO stability of CT LTI systems The I/O model of a LTI system with m inputs and p outputs can be described by an impulse response matrix, H : T Rp×m, whose elements →hij : T R represent the impulse response from input j to output i.→ ∞ yi (t) = hij (t − τ )uj (τ) dτ. −∞ Theorem A CT LTI system S with impulse response matrix H is BIBO stable if and only if �S�∞,ind = max 1≤i≤p m� j=1 � +∞ −∞ |hij (t)| dt < ∞. Note: in the scalar case (SISO), �S�∞,ind = �h�1, i.e., the ∞-induced norm of the system S is the L1 norm of the impulse response h (seen as a signal in the time domain). Often �S�∞,ind is referred to as the L1 norm of H in the general, MIMO case as well. E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 7 / 12������ ������� � �� � � Proof Sufficiency: m−∞ � j=1 +∞ hij (t − τ )uj (τ ) dτ�y�∞ = sup max t∈R 1≤i≤p m≤ sup max hij (t − τ )t∈R 1≤i≤pj=1 −∞ | | dτ · �u�∞ ≤ �S�∞,ind �u�∞ Necessity: Focus on the scalar case, i.e., R |h(t)|dt = ∞. +∞ Choose u such that u(t) = −sign(h(−t)). Clearly, �u�∞ ≤ 1. Then y(0) = R h(0 − τ)u(τ) dτ = R |h(τ )| dτ = ∞ E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 8 / 12Additional remarks A similar result holds in discrete time. For finite-dimensional LTI systems, one can construct a state-space model, and compute H(t) = CeAt B + Dδ(t), t ≥ 0, which has Laplace transform H(s) = C (sI − A)−1B + D. The system is BIBO stable if and only if the poles of H(s) are in the open left half plane. Asymptotic stability implies BIBO stability, but not viceversa. For LTI systems, BIBO stability implies p-stability for any p. For time-varying and nonlinear systems, the statements above do not necessarily hold. E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 9 / 12� L2-induced norm Theorem (H∞ norm is the L2-induced norm) The L2-induced norm of a causal, CT, LTI, stable system S with impulse response H(t) and transfer function H(s) is �S�2,ind = sup ω∈R σmax[H(jω)] = �H�∞. From Parseval’s equality, �y�2 = 1 � +∞ Y �(jω)Y (jω) dω. 22π −∞ Hence, 12 2σmax(H(jω))2U�(jω)U(jω) dω ≤ sup σmax[H(jω)]22.�y�2 ≤ 2π R ω �u�To show the bound is tight, pick (SISO case) u(t) = exp(�t + jω0t), i.e., U(s) = 1/(s − � − jω0), with � < 0. Then, �y �2 = |H(� + jω0)|2�u�2 2 2 As � 0, by the continuity of H on the imaginary axis, the gain approaches → |H(jω0)|. E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 10 / 12� �Computation of H∞ norm Theorem Let H(s) = C (sI − A)−1B be the transfer function of a stable, strictly causal (D = 0) LTI system. Define Mγ = � A 1 γ BBT − 1 γ C T C −AT � . Then �H�∞ < γ if and only if Mγ has no purely imaginary eigenvalues. �H� < γ if and only if I − 1 H�(jω)H(jω) is invertible for all ω ∈ R, i.e., if γ2 and only if Gγ (s) = I − γ12 HT (−s)H(s) −1 has no poles on the imaginary axis. The next step is to build a realization of Gγ (s). E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 11 / 12� � Computation of H∞ norm diagram with H(s) and HT (−s) in unit positive feedback HT (−s) = −BT (sI + A)−T C T , so a realization of this is (−AT , −C T , BT , 0). Putting together the realizations, and eliminating the internal variables, one gets the system matrix of the realization we seek as A 1 BBT Mγ = γ , −C T C −AT which proves the claim. E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 12 / 12MIT OpenCourseWare http://ocw.mit.edu 6.241J / 16.338J Dynamic Systems and Control Spring 2011 For information about