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Structured Stochastic Uncertainty Structured Stochastic Uncertainty Bassam Bamieh Department of Mechanical Engineering University of California at Santa Barbara Santa Barbara, CA, 93106 [email protected] Abstract— We consider linear time invariant systems in over the time interval Z+. This is done in contrast with the feedback with structured stochastic uncertainties. This setting standard setting over Z since stability arguments involve the encompasses linear systems with both additive and multiplica- growth of signals starting from some initial time. tive noise. We provide a purely input-output treatment of these systems without recourse to state space models, and For any random variable (or vector) v, we use σ := v⇤v to denote its variance, and ⌃ := vv⇤ thus our results are applicable to certain classes of distributed v E{ } v E{ } systems. We derive necessary and sufficient conditions for mean to denote its covariance matrix. A stochastic process u is + square stability in terms of the spectral radius of a linear a one-sided sequence of random variables uk; k Z . matrix operator whose dimension is that of the number of { 2 } We will thus denote by σ := u⇤u the sequence uncertainties, rather than the dimension of any underlying state uk E{ k k} of its variances, and by ⌃ = u u⇤ the sequence space models. Our condition includes the case of correlated uk E{ k k} uncertainties, and reproduces earlier results for uncorrelated of its inter-component correlation matrices. A process u is uncertainties. termed second order if it has finite covariances ⌃uk for each k Z+. Although the processes we consider are 2 I. INTRODUCTION technically not stationary (stationary processes are defined The setting we consider is that of a discrete-time Linear over the doubly infinite time axis), it can be shown that they Time Invariant (LTI) system in feedback with gains δ , are asymptotically stationary in the sense that their statistics G 1 ..., δn (see Figure 3). These gains are random processes that become approximately stationary in the limit of large time, are temporally independent, but possibly mutually correlated. or quasi-stationary in the terminology of [13]. This fact is The setting of LTI systems in feedback with structured not used in our treatment here and the preceding comment uncertainties is common in the robust controls literature is only included for clarification. where the uncertainties are typically norm-bounded opera- tors, real or complex deterministic gains [1]–[8]. The setting A. Input-output definition of mean square stability where the uncertainties are stochastic has been relatively less Let be a linear time invariant (MIMO) system. The studied [9]–[12], but it is well known that the necessary G system is completely characterized by its impulse response and sufficient condition for mean square stability in the G + which is a matrix valued sequence Gk; k Z . The presence of structured stochastic uncertainties is a bound { 2 } action of on an input signal u to produce an output signal on the spectral radius of a matrix of H2 norms of all the G y is given by the convolution sum subsystems of . G Our aim is to provide a rather elementary and purely k input-output treatment and derivation of the necessary and yk = Gk l ul. (1) − sufficient condition for mean square stability. In the process, l=0 X we define a new object, a linear matrix operator, which If the input u is a second order stochastic process, then captures how a feedback system amplifies covariances of it is clear from (1) that y has finite variance for any k, signals in a loop. A pleasant side effect is that the conditions k even in the cases where this variance may be unbouded in in the case of correlated uncertainties (which have been time. This leads to the following input-output definition of unknown) are almost as easy to state as the ones for un- Mean-Square Stability. correlated uncertainties. Those earlier results on uncorrelated Definition 1: The linear time invariant system is called uncertainties are easy to reproduce from the conditions we G provide. Mean-Square Stable (MSS) if for each second order white input process u with uniformly bounded variance, the cor- II. PRELIMINARIES responding output process y = u has uniformly bounded G variance All the signals we consider are defined on the half-infinite, discrete-time interval Z+ =[0, ) Z. The dynamical 1 ⇢ σy := y⇤yk M sup σu , (2) systems we consider are maps between various signal spaces k E{ k } k ✓ k ◆ Work supported by AFOSR award AFOSR FA9550-10-1-0143 where M is a constant independent of k and the process u. In this paper we deal exclusively with this kind of stability, where δ is the Kroneker delta function, and ⌃uk is the time and we therefore refer to MSS stable systems as simply varying, inter-component correlation matrix of uk. By think- stable. ing of the vector component index as a “spatial” variable, A standard calculation shows that we refer to such signals as temporally white and spatially k correlated. 2 σyk = Gl σuk l (3) A standard calculation then shows that for y and u related | | − l=0 by (1), their instantaneous correlations matrices are related X when u is a white process. A uniform bound can be deduced by k from the following inequality ykyk⇤ =: ⌃yk = Gk l ⌃ul Gk⇤ l. (6) E{ } − − l=0 1 2 X sup σyk Gl sup σuk . (4) k | | ! k C. Multiplication of processes Xl=0 ✓ ◆ Such quantities will occur often in the sequel, so we adopt Given any two vector-valued stationary stochastic pro- the notation cesses u and h, their element-by-element product process σy := sup σyk , y1(k) h1(k) u1(k) k k1 k . and note that the bound (4) can be rewritten in terms of the 2 . 3 := 2 . 3 2 y (k) h (k) u (k) H norm of as 6 n 7 6 n n 7 G 4 5 4 5 2 is also a stationary process. σy 2 σu . (5) k k1 kGk k k1 It is easy to see that equality holds when u has constant III. SISO UNSTRUCTURED UNCERTAINTY variance. Conversely, if does not have finite H2 norm, G We now consider the simplest case of uncertainty analysis equation (3) shows that any input with constant variance depicted in Figure 2. is a strictly causal LTI system, d G causes σyk to grow unboundedly, and thus the bound (2) will not hold for any finite M. In summary, we can conclude that a linear time invariant d u y system is MSS if and only if it has finite H2 norm, and in G G that case the inequality (5) holds, with equality in the case of the input having equal (in time) variance. z e w For a feedback interconnection, we define the MSS sta- δ bility of the overall system in a manner parallel to the con- ventional scheme of injecting exogenous disturbance signals into all loops. Consider the feedback system 1 with d1 and Fig. 2. LTI system in feedback with unstructured stochastic uncertainty d being white second order processes, and and are 2 G1 G2 linear causal systems. We say that the feedback system is and w are exogenous white processes with uniform variance, MSS if all signals u1, u2, y1 and y2 have finite variance and δ is a white process with uniform variance σδ and uniformly in time. independent of the signals d and w. We assume to have G finite H2 norm. The stability conditions we derive follow from a type of “small gain” analysis of the feedback interconnection in u G1 y d1 1 1 Figure 2 based on the variances of the signals in the loop. We therefore begin by deriving some basic relations between y2 u2 d2 those variances. 2 An important consequence of the independence of δ and G the exogenous signals is that the δ block “whitens” its input signal e, i.e. even though e will in general be colored, z is white. This can be seen by letting k2 >k1 and calculating Fig. 1. MSS stability for a feedback interconnection z z = δ e δ e E{ k1 k2 } E{ k1 k1 k2 k2 } = δ e e δ =0, B. The MIMO case E{ k1 k1 k2 }E{ k2 } When is a MIMO system, the convolution input-output due to the independence of δk of the other signals. In fact, a G 2 description (1) still holds with u and y as vector signals and consequence of the strict causality assumption is that current G a sequence of matrices. Consider input signals that are and future values δ, d and w are independent of past values { k} temporally white but with correlated components such as of any of the internal signals. When k1 = k2, we get 2 2 2 uk1 uk⇤ = δk1 k2 ⌃uk , σzk = ek = δk ek = σδ σek . (7) E 2 − 1 E E Similarly, the signal u is also white as can be seen from the Consider any time horizon k¯, and note that the quantity k¯ 2 following calculation with k2 >k1 ↵ := G can be made arbitrarily close to 1. l=0 l kGk2 ≥ The monotonicity of the sequence σu gives the following uk1 uk2 = (dk1 + zk1 )(dk2 + zk2 ) P E{ } E{ } lower bounds = dk zk + dk zk E{ 1 2 } E{ 2 1 } nk¯ 2 = dk1 δk2 ek2 +0 σ G σ + σ E{ } unk¯ nk¯ l ul d ≥ − = d e δ =0. l=0 E{ k1 k2 }E{ k2 } X nk¯ For the case of k2 = k1 we get 2 Gnk¯ lσul + σd ≥ − σ = σ + σ = σ + σ σ .
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