6.241 Dynamic Systems and Control Lecture 22: Balanced Realization Readings: DDV, Chapter 26 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology April 27, 2011 E. Frazzoli (MIT) Lecture 22: Balanced Realization April 27, 2011 1 / 10Minimal Realizations We have seen in the previous lectures how to obtain minimal realizations from non-minimal realizations (i.e., keeping the reachable and observable part from the Kalman decomposition), and also algorithms to construct minimal realizations of a transfer functions. Minimal realizations are unique up to similarity transformations. However, there are some realizations that are more useful than others, for a number of reasons Kalman decomposition Standard forms Canonical forms . . . In this lecture we will consider what is known as balanced realization. E. Frazzoli (MIT) Lecture 22: Balanced Realization April 27, 2011 2 / 10The Hankel Operator Consider for simplicity a discrete-time system G with state-space realization (A, B, C , D), and transfer function H(z), with impulse response (H0, H1, H2, . . .). How do outputs at time steps k ≥ 0 depend on inputs at time steps k < 0? ⎤⎡ H0 H1 H2 · · · ⎤⎡ ⎤⎡ y[0] u[−1] u[−2]⎥⎥⎥⎥⎦ ⎢⎢⎢⎢⎣ H1 H2⎢⎢⎢⎣ ⎥⎥⎥⎦ ⎢⎢⎢⎣ ⎥⎥⎥⎦ y[1] · · · H2 . . . . . . · · · · · · = Huy+ = = y[2] u[−3] −, . . . . . . . . . . . . . . . . . . the Hankel operator H transforms past inputs u− into future outputs y+. E. Frazzoli (MIT) Lecture 22: Balanced Realization April 27, 2011 3 / 10� � Structure of the Hankel Operator Recall that H0 = D, and Hk = CAk−1B. The Hankel operator can be written as ⎤⎡⎤⎡H0 H1 H2 · · · C ⎢⎢⎢⎢⎣ ⎥⎥⎥⎥⎦ = ⎢⎢⎢⎣ CA CA2 H1 H2⎥⎥⎥⎦ · · · · · · B AB A2B = O∞R· · · H = .. H2 .. .. · · · ∞ . . . .. . . . . . . . . . . Since (A, B, C , D) is a minimal realization, Rank(H) = n. In particular, H will have exactly n non-zero singular values, which are also called the Hankel singular values of the system G . E. Frazzoli (MIT) Lecture 22: Balanced Realization April 27, 2011 4 / 10� � Computation of the Hankel singular values Recall that, given the properties of singular values, σi (H) = λi (HHT ) = λi (HT H). Notice that T = O∞R∞RT OT = O∞POTHH∞ ∞ ∞ T T TThe (DT) reachability Gramian P R R satisfies APA P BB− −= = .∞T T T TO O PO QPO= = ∞∞Similarly, Q = OT O∞, and AT QA − Q = ∞ −C T C .∞Since HHT wi = σi 2wi by definition, we also have wi∞∞∞T 2 TOσ= i ∞OHH wi wi wi . In other words, σi (H) = σi (PQ), i = 1, . . . , n. the Hankel singular values can be easily computed from the knowledge of the reachability and observability Gramians. E. Frazzoli (MIT) Lecture 22: Balanced Realization April 27, 2011 5 / 10Hankel norm of a system Consider bounded-energy “past” input signals �u−�2 < ∞. How much does the energy of the past input get amplified in the energy of the “future” output signal �y+�2? This is an induced 2-norm, called the Hankel norm: �G �H := sup �y+�2 . =0�u−�2��u−�2 This can be computed easily as �G �H = σmax(H) = σmax(PQ). Note that, for any system G , �G �H ≤ �G �∞. The state x[0], depending on the realization, separates past and future: The energy necessary to drive the system to x[0] (i.e., �u−�2) is determined by (the inverse of) the reachability Gramian P. The energy in the output from x[0] (i.e., �y+�2) is determined by the observability Gramian Q. (Note that �y+�22 = x[0]T CT Cx[0] + x [0]T AT C T CAx[0] + . . . = x[0]T Qx[0], similarly for the control effort.) E. Frazzoli (MIT) Lecture 22: Balanced Realization April 27, 2011 6 / 10Balanced Realization It is of interest to “balance” the energy allocation between past control effort and future output energy, i.e., to equalize P and Q. A balanced realization is such that P = Q = diag(σ1, σ2, . . .). Can we find a similarity transformation T such that the realization is balanced? Recall (A, B, C , D) (T −1AT , T −1B, CT , D).→ Gramians are transformed as APAT − P = −BBT → T −1PT T AT T −T − Pˆ= −T −1BBT −T ,AT ˆ i.e., P T −1PT −T = Pˆ. Similarly, Q T T QT = Qˆ. → → E. Frazzoli (MIT) Lecture 22: Balanced Realization April 27, 2011 7 / 10Balanced Realization We would like PˆQˆ= diag(σ12, σ22, . . . , σn2) = Σ2 . In other words, T −1PT −T T T QT = T −1PQT = Σ2 . Since Q is positive definite, one can find a matrix R such that Q = RT R. Hence, T −1PRT RT = (RT )−1RPRT (RT ) = Σ2 RPRT is symmetric and positive definite, and can be diagonalized by an orthogonal matrix U, such that RPRT = UΣ2UT . Choose T = R−1UΣ1/2; then, Pˆ= Σ−1/2UT RPRT UΣ−1/2 = Σ, and similarly for Σ. E. Frazzoli (MIT) Lecture 22: Balanced Realization April 27, 2011 8 / 10Model Reduction Assume that we have a stable system G , with a minimal realization of order n >> 1. It is desired to find a reduced-order model (of order k < n) in such a way that some “error” is reduced. A possible criterion is to find the reduced-order model that minimizes the Hankel norm of the error, i.e., such that �G − G k �H is minimized. Clearly �G − G k �H ≥ σk+1(H). It is possible to compute a model that achieves exactly this bound (Glover ’84), but the procedure will not be covered in this course (see, e.g., 6.242). E. Frazzoli (MIT) Lecture 22: Balanced Realization April 27, 2011 9 / 10� Model reduction through balanced truncation A commonly used procedure for model reduction is based on the balanced realization. Idea: remove from the system matrices (in the balanced realization) the blocks corresponding to the smaller Hankel singular values. ⎡ ⎤ � � A11 A12 B1 � � Σ = Σ1 0 G : ⎣A21 A22 B2⎦ G k : A11 B1 0 Σ2 → → C1 D C1 C2 D If Σ1 and Σ2 do not contain any common elements, then the two resulting systems (in particular, the reduced-order model) will be stable. We have the following bounds: σk+1(H) ≤ �G − G k �H ≤ �G − G k �∞ ≤ 2 σl (H). l >k E. Frazzoli (MIT) Lecture 22: Balanced Realization April 27, 2011 10 / 10MIT OpenCourseWare http://ocw.mit.edu 6.241J / 16.338J Dynamic Systems and Control Spring 2011 For information about citing these materials or our Terms of Use, visit: