Stabilization with Finite Data Rates in Control Systems, Study Guides, Projects, Research of Electrical and Electronics Engineering

Download Stabilization with Finite Data Rates in Control Systems and more Study Guides, Projects, Research Electrical and Electronics Engineering in PDF only on Docsity! EECS 558 Term Project Stabilization with Finite Data Rates Dinesh Krithivasan Ramji Venkataramanan 1 Introduction In this term project, we study the effects of finite communication rates in control problems. Tradi- tionally, the communication channel between the plant and the controller is not modelled. This is because it is assumed that the outputs of the former (noisy or noiseless) are available completely and with infinite precision to the latter. In this work, we study systems where the link between the plant and the controller can carry only a limited number of data symbols per unit time. We refer to these as systems with finite data rate. We consider three different systems in the order of increasing complexity. The first [1] is a scalar, linear time varying system with finite data rate. The conditions under which this system is asymp- totically stabilizable are studied. The second system [2] is a finite dimensional, linear time invariant system. The exponential stabilizability of this system under a data rate constraint is studied. Finally, we consider the most general case - a finite dimensional, linear system with both process and obser- vation noise [3]. We find conditions that the data rate should satisfy in order to ensure mean square state stability of this system. Before we proceed, a word about the notation used. We use upper case letter to denote random vari- ables, lower-case ones to denote their realizations. A sequence [x0, x1, . . . , xk] is denoted xk. Matrices and vectors are denoted using bold font. 1 Controller CoderDelay 1 U X k k k-1S Plant: Xk Sk Figure 1: Scalar system with limited data rate. 2 Scalar Linear Time-Varying System We consider the following discrete-time control system. xk+1 = akxk + uk, ∀k > 0, (1) where xk and uk are the system state and control signal, respectively, at time k. We also assume that the initial state X0 is chosen randomly according to a distribution function P such that E|X0|m < ∞ for some m > 0. Further, suppose there is a coder that observes the states noiselessly and then sends data to the controller over a noiseless, rate-limited channel. The channel is such that it can carry only one symbol sk from a fixed alphabet Zµ , {0, 1, . . . , µ− 1} during each time step. The corresponding finite data rate is then R , log2 µ bits per time step. The system is shown in Figure 1. As shown, the symbol sk takes one time step to reach the controller. This means that at time k, the controller has sk−1 available and produces a control action uk = gk(sk−1), ∀k > 0. (2) The symbol sk transmitted by the coder at time k is a function of the past and present states xk and past symbols sk−1. However, (1) and (2) imply that xk is completely determined by sk−2, which determine the control actions up to time k − 1 and x0. Hence, sk is a function of x0 and sk−1. By 2 2.2 Infinite Horizon The infinite horizon problem is to find conditions under which the system is asymptotically stabilizable in the sense of E|Xk|m → 0 as k →∞. (8) The optimal coder-controller is similar to the finite horizon case. Recall that in the finite horizon case, the coder computes the normalized value of the quantizer index and sends it to the controller, one µ−ary digit at a time. In the infinite horizon case, we can exploit the fact that a closed form expression exists for the normalized density of quantizer points of an MmPE-optimal quantizer. Denote this density by ν. Then, the normalized quantizer index for x0 is given by c(x0) , ∫ y≤x0 ν(y)dy, ∀x0 ∈ R. (9) This leads to the following scheme in the infinite horizon case. • Coder:For k = 0, 1, . . ., the symbol sk is the (k + 1)−th digit in the µ− ary expansion of c(x0), given by (9). • Controller: At time k, calculate ηk(sk−1) = c−1(ĉk−1), (10) and use (6) to generate the control signal. A necessary and sufficient for the plant (1) is now discussed. Using a result from asymptotic quantization theory, it is shown that this coder- controller pair is optimal. Theorem 1 The system described in (1) is asymptotically stabilizable in the sense that there exists a coder-controller scheme that takes E|Xk|m → 0 if and only if lim k→∞ µ−k k−1∏ j=0 |aj | = 0 (11) Further, the scheme described above is optimal in this sense with minimal cost J∗m,∞ = (m + 1) −12−m‖p‖1/(m+1) (12) 5 Controller CoderDelay 1 U CX k k k-1S Plant: Xk Sk Figure 2: Finite Dimensional LTI system with limited data rate. Condition (11) can be interpreted as comparing the speed with which the initial condition estimate becomes accurate and that with which the state of the open-loop system changes. 3 Finite Dimensional Linear Time-Invariant System In this section, we generalize the plant model presented in the previous section to allow the state values to be vector valued. Consider the discrete-time, linear time-invariant system shown in figure 2. It can be described by the following set of equations. xk+1 = Axk + Buk (13) yk = Cxk ∀k ∈ Z+ where xk ∈ Rn is the state, yk ∈ Rp is the sensor measurement and uk ∈ Rm is the control vector at time k. The following assumptions are made about the system. • (A,B) is reachable and (C,A) is observable. • The initial state x0 is a realization of a random variable X0 which has a density px0 and satisfies E ‖X0‖r+ < ∞ for some r,  > 0. The sensor is connected to the controller through a noiseless digital channel that can carry one symbol sk from the alphabet Zµ = {0, 1, . . . , µ−1} per sampling interval. The rate of this code is R , 6 log2 µ bits per interval. In the spirit of source coding, issues such as finite memory and computational complexity of the encoder will be neglected to focus attention on the communication aspect of the problem. Delay between the coder and controller is also neglected. Under these assumptions, sk = γk(yk, sk−1) (14) where γk : Rp×(k+1)×Zµ :→ Zµ is the encoder mapping at time k. At time k, the controller has access to the symbols s0, . . . , sk−1 and generates the signal uk = gk(sk−1) (15) where gk : Zkµ → Rm is the controller mapping at time k. Given a rate R > 0, we are interested in finding whether there exists a coder-controller pair (γ, g) which ρ-exponentially stabilizes the plant (13) in the following sense. ρ−krE‖ Xk ‖r →∞ as k →∞ (16) The main result of this section is Theorem 2 Assume that the LTI plant (13) is controllable and observable with (possibly repeated and conjugate) eigenvalues η1, . . . , ηn. Further, assume that its initial state x0 ∈ Rn is random with a distribution that is absolutely continuous with respect to Lebesgue measure on Rn and finite (r + )th moment E‖X0‖r+ < ∞ for some r,  > 0. Then for a given data rate R (bits per sampling interval), a coder-controller (14)-(15) that ρ-exponentially stabilizes the system in the sense of (16) if and only if R > ∑ |ηj |≥ρ log2 ∣∣∣∣ηjρ ∣∣∣∣ (17) The expression for rate can be rewritten as µ > ∏ |ηj |≥ρ | ηj ρ |. The left side of (17) gives the number of disjoint regions into which a µ-point quantizer can divide it. The right hand side of (17) gives the rate at which the volume of the state subspace corresponding to eigenvalues with |ηj | ≥ ρ increases at each time step. 7 The quantity ‖pX‖ 2θ n(1−θ) θ is known in information theory as the Rényi differential entropy power of order θ and is a measure of the radius squared of the effective support volume of the density. Using lemma (1) to bound the expectation in (25), we get E‖(ρ−1J)kX́0 − ρ−kqk−1(X́0)‖r ≥ β µ (k−1)r n (∫ p(ρ−1J)kX́0(x) θdη(x) ) r n(1−θ) (27) Changing the variable of integration to y = (ρJ−1)kx, the above integral evaluates to E‖(ρ−1J)kX́0 − ρ−kqk−1(X́0)‖r ≥ β µ (k−1)r n ( ρ−n ∏ i |ηi|ni ) kr n ‖pX́0‖ rθ n(1−θ) θ (28) For ρ-exponential stability we want the RHS of (28) →∞ as k →∞. As ‖pX́0‖θ > 0, it is necessary that 2R = µ > ∏ i | ηi ρ | ni ⇔ R > ∑ j:|ηj |≥ρ log2 | ηj ρ |. This proves the necessity part of theorem (2) 3.3 Proof of Achievability We now build a specific coder-controller that can ρ-exponentially stabilize the system (13) for any given rate satisfying (17). We will once again assume without loss of generality that all the dynamical modes have ηi ≥ ρ. The basic idea is to recursively quantize the initial state. By means of a time-sharing protocol, each scalar component of the i-th subsystem is allocated a data rate roughly proportional to log2 | ηi ρ | which is used to quantize and transmit it with increasing accuracy. The construction is essentially the same as that of the coder-controller designed for the problem discussed in section 1. We build a coder-controller similar to those described by (9) and (10) for each of the ν subsystems. The coder and controller are then described as follows • Coder: Divide times k ≥ n into epochs [n + jτ, . . . , n + jτ + τ − 1], j ∈ Z+ of uniform integer duration τ ≥ n and subdivide each epoch into ν subepochs of duration niτi where τi , dτR−1 log2 | ηi ρ |e (29) Calculate the transformed initial state x́0 by solving W0T−1x́0 = [yT0 , . . . ,y T n−1] 10 where the observability matrix W0 , [ CT (CA)T , . . . , (CAn−1) T ]T has rank n. During the i-th subepoch of the j-th epoch, transmit the (j + 1)th block of τi successive bits in the µ-ary expansion of c(x́(i)0 ). Here, the normalized quantizer index c(.) is defined similar to (9) • Controller: Set u0, . . . ,un+τ−1 = 0. At each time instant, estimate the transformed initial state component x́(i)0 for all the subsystems by inverting the compressor c(.) similar to (10). Then calculate the control signals from this estimate of the initial state. Note that this is possible because of the reachability of (J,TB). Analysis of this scheme uses ideas from asymptotic quantization theory which provides upper bounds on the r-th state absolute moments generated by this coder-controller. It can be shown that for the coder-controller scheme described above, as long as the time sharing scheme described by (29) is feasible, the resulting system is ρ-exponentially stabilized. For the protocol (29) to be feasible, it needs to satisfy ∑ i niτi ≤ ∑ i ni ( τ log2 |ηi/ρ| R + 1 ) = τ ∑ i ni log2 |ηi/ρ| R + ∑ i ni = τ ∑ j:|ηj≥ρ log2 |ηj/ρ| R + n, ∀τ ∈ N This shows that for sufficiently large τ and any rate R above the critical rate given by theorem (2), the time sharing protocol (29) is feasible and hence the system is exponentially stabilizable. 4 Finite Dimensional Stochastic Linear System In this section, we consider a generalization of the plant described in the previous section by allowing the presence of noise in both the process and the measurement. Consider the partially-observed, discrete-time, stochastic linear system shown in figure 3. It can 11 Controller CoderDelay d U Y k k k-dS Plant: Xk Sk Figure 3: Scalar system with limited data rate. be described by the following set of equations. xk+1 = Axk + Buk + vk (30) yk = Cxk + wk ∀k ∈ W with state xk and process noise vk ∈ Rn, control signal uk ∈ Rm and measurement yk and measurement noise wk ∈ Rp. The following assumptions are made about the system. • (A,B) is reachable and (C,A) observable. • The basic random variables X0,Vk,Wj are mutually independent ∀k, j ∈ W • ∃ > 0 s.t the basic random variables X0,Vk,Wj have uniformly bounded (2 + )-th absolute moments over k ∈ W • The probability distribution of each random variable Vk is absolutely continuous with respect to Lebesgue measure λ on Rn • infk∈W H{Vuk} > −∞ where Vuk ∈ Rf×n is the process noise seen by the f ≥ 1 unstable eigenvectors of A. In words, the process noise injects a minimum amount of uncertainty into the unstable dynamics. The sensor that observes the noisy measurements Yk is connected to the controller through a noiseless digital channel onto which onto which one symbol sk from a finite alphabet Sk, of possibly 12 equal to the critical bound H. More importantly, it does not tell us the behaviour of the mean square state norm as R ↘ H. These aspects are better illustrated in the entropy-based analysis below. The conditional entropy power of a random variable X ∈ Rf , given an event A = a is defined as Na{X} , (2πe)−1e2Ha{X}/f (37) This serves a purpose in the proof similar to that of Renyi differential entropy of section 2. Na({X})f/2 can be regarded as the volume of the effective support set of pX|a. Furthermore, Na({X}) ≤ e 1 f −1 Ea‖X‖2 (38) with equality iff X is symmetric Gaussian with zero mean when conditioned on A = a. Another important property of the conditional entropy power is its superadditivity which can be expressed as Na{X + Y} ≥ Na{X}+ Na{Y} (39) when X,Y ∈ Rf are mutually independent when conditioned on the event A = a. We will derive a recursion for nk , N{Xk|Sk−d−1}. We have nk = N{Xk|Sk−d−1} (40) = E{NSk−d−1{Xk}} ≤ e1/f−1E{ESk−d−1‖Xk‖2} = e1/f−1E‖Xk‖2 ∀k ∈ W So, boundedness of {nk}k∈W is a necessary condition for (34) to hold. The dynamical equation for xk+1 can be written using (33) as xk+1 = Jxk + TBgk(sk−d) + Tvk 15 The conditional entropy power of xk+1 can then be written as NSk−d{Xk+1} = NSk−d{JXk + TVk + TBgk(Sk−d)} = NSk−d{JXk + TVk} ≥ NSk−d{JXk}+ NSk−d{TVk} by superadditivity = NSk−d{JXk}+ N{TVk} = |detJ|2/nNSk−d{Xk}+ N{TVk} ∀k ∈ W (41) Taking expectation on both sides and using the definition (40), we can derive a recursion for nk as nk+1 ≥ | det J µk−d |nk + β (42) where β , N{TVj} is strictly positive by assumption. Further manipulation of the recursion gives us log2 ( 1− 11 β supk∈W nk ) ≥ 2 n (log2 |det J| −R) = 2 n (H −R) This proves the strict necessity of (35). Rearranging this inequality tells us the behaviour of the mean square state norm. sup k∈Z+ E‖Xk‖2 ≥ βe1−1/n 1− 2−2(R−H)/n (43) β in (43) depends only on the noise statistics and hence this equation gives a universal bound on the performance of all coder-controllers at a given rate R. This clearly shows that performance drastically reduces as R ↘ H. 4.3 Proof of Achievability In this section, we show the achievability of all data rates that satisfy (35) by explicit construction of coder-controllers. The chief complications in the design and analysis of the scheme presented arise from the possibly unbounded support of the noise terms. With uniformly bounded noise, any coding and control law that achieves asymptotic contraction without disturbances can be modified to ensure 16 uniform bound on the state norm. The idea is that bounded disturbances boost the volume of the worst-case region in which the state lies by an additive constant and hence can be negated easily. The semi-heuristic coder-controller design presented below is inspired by the optimal coding and control schemes in the case of a linear Gauss-Markov system with mean quadratic cost. The structure of the coder and controller are described below. • Coder: Prior to coding, a Kalman filter is applied recursively to calculate the linear minimum variance prediction xk+d|k of x́k+d given the measurement sequence yk and control sequence uk+d−1. Note that the control signals are not observed directly by the coder but are inferred from knowledge of the symbol sequence sk−1 and the controller mappings. • Based on the past symbol sequence sk−1, the latest prediction xk+d is recursively and possibly non-uniformly quantized to yield a coded estimate x̂k+d = Qk(xk+d, sk−1) with µk possible values. • Upon receiving sk at time (k + d), the controller uses the sequence sk o regenerate x̂k+d and applies a certainty-equivalent linear control law uk+d = Lx̂k+d Several notes regarding the structure of this control-controller scheme are in order. Although no Gaussian assumptions have been made, note that the linear minimum variance pre- dictions satisfy the separation principle E‖X́k‖2 = E‖X́k −Xk|k−d‖2 + E‖Xk|k−d‖2 ∀k ∈ W even with non-Gaussian noise. The first term in this expression is uniformly bounded (by observability) and hence the mean square stability of X́k and Xk|k−d are equivalent. Since the objective here is not optimality but a weak notion of stability, it is natural to investigate whether simple quantizers will suffice. It turns out that if the plant is strictly unstable, using any static memoryless coder or finite-state predictive quantizers will result in absolute state moments unbounded 17