Stabilization of Networked Control Systems with Clock Offset

Transcription

1 Stabilization of Networked Control Systems with Clock Offset Masashi Wakaiki, Kunihisa Okano, and João P Hespanha Abstract We consider the stabilization of networked control systems with time-invariant clock offset between the sensors and the controllers Clock offsets are modeled as parametric uncertainty and we provide necessary and sufficient conditions for the existence of a single controller that is capable of stabilizing the closed loop for every parameter value in a given range For scalar systems, using the Nevanlinna-Pick interpolation, we obtain the maximum length of the offset interval in which the system can be stabilized by a single linear time-invariant controller I INTRODUCTION In networked control systems, real-time data on plants is not available at the controller side due to transmission and/or computation delays Since these delays are unknown, in general, the sensors transmit data with time-stamps to provide information on the time at which the data was collected The problem is that the clocks in the sensors and at the controller may differ Hence protocols to establish synchronization have been actively studied as surveyed in [1], [2], and synchronization using Global Positioning Systems (GPS) or radio clocks have been utilized in practical systems However, synchronizing clocks over networks has fundamental limits [3] Moreover, recent works [4] [6] have shown that synchronization technologies based on GPS signals may be vulnerable against malicious attacks In this paper, we consider a networked control system in which the internal clock of the sensor is not synchronized with that of the controller Our objective is to determine how large the error between the clocks can be without compromising the existence of a single linear time-invariant controller that stabilizes the closed loop We formulate the feedback stabilization with clock offset as the problem of stabilizing systems with parameter uncertainty and provide a necessary and sufficient condition for the existence of a single controller that stabilizes the closed loop for every value of the parameter in a given range For general-order systems, from this condition we can only obtain computationally efficient conditions that are either necessary or sufficient For the special case of scalar systems, the condition leads to the exact bound on the clock offset that can be allowed for stability We see that the bound varies depending on the class of stabilizers Larger offsets can be tolerable under the class of linear time-invariant stabilizers This material is based upon work supported by The Kyoto University Foundation, JSPS Postdoctoral Fellowships for Research Abroad, and the National Science Foundation under Grant No CNS The authors are with the Center for Control, Dynamicalsystems and Computation (CCDC), University of California, Santa Barbara, CA USA ( mwakaiki@eceucsbedu; kokano@eceucsbedu; hespanha@eceucsbedu) compared with the cases of static stabilizers and 2-periodic static stabilizers The derivation for the main results is based on algebraic approaches to the problem of simultaneous stabilization [7] and rely on the parity interlacing property [8], [9, Section 54] However, we had to overcome a few technical difficulties that distinguish the problem considered here from previously published results: (i) Infinitely many plants: We consider a family of process models that is indexed by a continuous-valued parameter Such a family includes infinitely many plants On the other hand, the simultaneous stabilization of three plants is rationally undeciable [10] Also, many techniques for simultaneous stabilization are based on linear matrix inequalities (LMIs); see, eg, [11], [12] and the references therein The LMI-based approach exploits the property that the number of plants is finite, and hence we cannot use it in this work (ii) Common unstable poles and zeros: The earlier studies on simultaneous stabilization consider a restricted class of plants For example, it is assumed in [13], [14] that plants have no unstable zeros Similarly, in [15], a sufficient condition is obtained for a family of plants with no common unstable zeros or poles The set of plants in [16], [17] has common unstable zeros (or poles) but all of the plants are stable (or minimum-phase) These assumptions are not satisfied for the systems in the present paper The systems we consider have common unstable poles and zeros, and furthermore some of them have other unstable zeros The remainder of the paper is organized as follows Section II gives the systems we stabilize and presents the problem formulation In Section III, we consider generalorder systems and derive a necessary condition and a sufficient condition for the stabilization problem with clock offset For scalar systems, we show the exact bound on the permissible clock offset in Section IV Section V is devoted to the comparison of the bound under general linear time-invariant stabilizers with that under specific classes of stabilizers for scalar systems Notation and definitions: Let D, D, and T denote the open unit disc {z C : z < 1}, the closed unit disc {z C : z 1}, and the unit circle {z C : z = 1}, respectively We denote by H the space of all bounded holomorphic scalar-valued functions in D RH denotes the subspace of H consisting of all stable real-rational functions Let us denote the field of fraction of H by F For a commutative ring R, M(R) denotes the set of matrices with entries in R, of whatever order For M C p q, let M be the induced 2-norm, which is equal to the largest

2 s k t k 1 =(k 1)h t k = kh s k+1 ŝ k ŝ k+1 Time Fig 1: Sampling instants s k, reported time-stamps ŝ k, and updating instants t k of the zero-order hold singular value of M For G M(H ), the H -norm is defined as G = sup z D G(z) We say that C M(F ) stabilizes P M(F ) if (I + P C) 1, C(I + P C) 1, and (I + P C) 1 P belong to M(H ), and we call C a stabilizer of P A pair (N, D) in M(H ) is said to be right coprime if the Bezout identity XN + Y D = I holds for some X, Y M(H ) P M(F ) admits a right coprime factorization if there exist D, N M(H ) such that P = ND 1 and the pair (N, D) is right coprime Similarly, a pair ( D, Ñ) in M(H ) is right coprime if the Bezout identity Ñ X + DỸ = I holds for some X, Ỹ M(H ) P M(F ) admits a left coprime factorization if there exist D, Ñ M(H ) such that P = Ñ D 1 and the pair ( D, Ñ) is left coprime If P is a scalar-valued function, we use the expressions coprime and coprime factorization II PROBLEM STATEMENT Consider the following plant: ẋ(t) = Ax(t) + Bu(t), (II1) where x(t) R n and u(t) R m are the state and the input of the plant, respectively Let s 1, s 2, be sampling instants The sensor observes the state x(s k ) and sends it to the controller together with a time-stamp However, since the sensor and the controller share no global clock, the time-stamp typically includes an unknown offset with respect to the controller clock In this paper, we assume that the offset is fixed, that is, the timestamp ŝ k reported by the sensor is given by ŝ k = s k + (k N) for some unknown constant R This means that there is a time-invariant offset between the sensor and the controller but no difference in their frequencies, or that this difference is negligeable for the time-scales of interest The control signal is assumed to be piecewise constant and updated periodically at times t k = kh (k N) with values u k computed by a controller: u(t) = u k for t [t k, t k+1 ) While the control updates are assumed periodic, the true sampling times s k and the reported sampling times ŝ k may not be periodic However, we do assume that both s k and ŝ k do not fall behind t k by more than h This assumption is formally stated as follows Assumption 21: Fix h > 0 For k N, t k t k 1 = h and s k, ŝ k [t k 1, t k ) Fig 1 shows the timing diagram of the sampling instants, the reported time-stamps, and updating instants of the control inputs u(t k ) Discre'zed extended system u(t) Stabilizer Plant ZOH Sampler Clock Offset ˆx(t k ) x(t) x(s k ) Es'mator s k ŝ k ˆx(ŝ k ) Fig 2: Closed-loop system with clock offset Using the received state x(s k ) and the reported time-stamp ŝ k, the controller can estimate the plant state in the following way: ˆx(t) = Aˆx(t) + Bu(t), ˆx(ŝ k ) = x(s k ) (k N), where ˆx R n is the estimated state This estimate leads to the discretized extended system in Fig 2, with state and input given by x(tk ) ˆx(t ξ k = k ), u ˆx(t k ) k = u(t k ), respectively, which evolves according to ξ k+1 = F ξ k + G u k, y k = H ξ k, (II2) where p := e Ah, := e A I, and p p F := p(i + ) p(i + ) [ ( h p G := 0 e Aτ dτ (I +) ) ] h e Aτ dτ B 0 p(i + ) h e Aτ dτb 0 H := [ 0 I ] (II3) The objective of the present paper is to obtain lower and upper bounds on the clock offset under which there exists a single linear time-invariant stabilizer of (II2) for every value of in the corresponding range To obtain these bounds, in what follows we consider the problem below Problem 22: Given an interval (, ), determine if there exists a single stabilizer of the extended system (II2) for every (, ) Our second assumption is stabilizability and detectability for the existence of stabilizers Assumption 23: For all (, ), (F, G, H ) is stabilizable and detectable Before we turn to the stabilization analysis, it is convenient to introduce another representation of the system (II2): If A is invertible, then G in (II3) is given by [ G = pa 1 B (p(i + ) I)A 1 B ] Note also that with the similarity transformation T given by I T :=, I + I

3 we have F := T 1 p 0 F T = 0 0 Ḡ := T 1 (p I)A G = 1 B A 1 B H := H T = [ I + I ] (II4) It follows that the eigenvalues of F are those of p in addition to n eigenvalues equal to zero, and hence the eigenvalues of F are independent of III ANALYSIS VIA SIMULTANEOUS STABILIZATION We first consider a general simultaneous stabilization problem not limited to the system introduced in Section II Consider the family of plants P M(RF ) parametrized by S, where S is a nonempty index set Assume that we have a doubly coprime factorization of P over RH : [ Y X D X ] = I, (III1) Ñ D N Ỹ where P = D 1 N 1 and P = Ñ D are a right coprime factorization and a left coprime factorization, respectively The following theorem gives a necessary and sufficient condition for simultaneous stabilization: Theorem 31 ([7], [9]): Let 0 S Define S + := S \ { 0 } and U Y0 X := 0 D ( S V Ñ 0 D0 N + ) (III2) Then (V, U ) is right coprime for every S + Moreover, there exists a stabilizer of P for every S if and only if there exists Q M(RH ) such that for all S +, Such a stabilizer is given by (U + QV ) 1 M(RH ) (III3) C := (Y 0 QÑ 0 ) 1 (X 0 + Q D 0 ) (III4) Remark 32: Most literature on simultaneous stabilization considers a family of finitely many plants However, the result in Theorem 31 holds for an arbitrary family; see the last paragraph of Section III in [7] It is often not easy to verify the existence of Q with (III3) in a computationally efficient fashion To address this challenge, in the remainder of this section we derive computationally attractive conditions that are either necessary or sufficient (but not both) Later in Section IV, we shall explore the specific structure of the system in (II3) to derive a condition that is both necessary and sufficient, but only applies to scalar systems However, for the rest of this section, we continue to consider a general process P with factorization in (III1) We start by providing a necessary condition that follows from a well-known result on the simultaneous stabilization of two processes via the parity interlacing property Corollary 33: Let 0, 1 S and define [ Y0 X 0 U1 := V 1 Ñ 0 D0 ] D1 N 1 If there exists a stabilizer of P for S, then for all 0, 1 S, V 1 U 1 1 satisfies the parity interlacing property, that is, the number of poles of V 1 U 1 1 (counted according to their McMillan degrees) between any pair of real blocking zeros of V 1 U 1 1 in D is even Proof: If V 1 U 1 1 does not satisfy the parity interlacing property, then P 1 and P 2 do not have a common stabilizer [9, Section 54] Hence there exists no stabilizer of P for all S The next result gives a sufficient condition for (III3) under certain assumptions on D and N For the system (II2), Proposition 36 facilitates verifying the assumptions of the theorem Theorem 34: Let 0 S Define S + := S \ { 0 } Assume that D = D 0 for all S, and that we can write N (z) N 0 (z) = L(z)f(), (III5) where L M(RF ) and f() M(R) for S Define γ := inf (X Q M(RH 0 + Q D 0 )L ) (III6) If f() < 1/γ for S, then C in (III4) stabilizes P for every S Proof: We define U and V as in (III2) If D = D 0, then the Bezout identity Y 0 D 0 + X 0 N 0 = I leads to U = I + (Y 0 (D D 0 ) + X 0 (N N 0 )) In addition, since V = D 0 ( = I + X 0 (N N 0 ) D 1 0 Ñ 0 = N 0 D 1, we obtain 1 D 0 Ñ 0 N D 1 )D = D 0 (N N 0 ) Hence U 0 + QV = I + (X 0 + Q 0 D0 )(N N 0 ) In conjunction with the small gain theorem, the above equation shows that if (X 0 + Q D 0 )(N N 0 ) < 1 ( S), then (III3) holds for all S From the assumption (III5), (X 0 +Q D 0 )L f() (X 0 +Q D 0 )(N N 0 ) Hence if f() < 1/γ for S, then P is simultaneously stabilizable by C in (III4) from Theorem 31 Remark 35: The minimization problem (III6) is solvable by reducing it to the Nehari problem We can therefore check the sufficient condition in Theorem 34 by efficient matrix computations; see [18, Chapter 8] for details The proposition below shows that the system (II1) always satisfies the assumptions on D and N in Theorem 34 if it is a single-input system with invertible A We also obtain L and f in (III5) without calculating a coprime factorization of P for all S Proposition 36: Suppose that 0 (, ) and that the system (II1) is a single-input system Define P (z) := zh (I zf ) 1 G For all (, ), there exists a right coprime factorization P = D 1 N satisfying D = D 0 Furthermore, suppose that A is invertible and

4 is in Jordan canonical form Since B is a vector and since e A is upper triangular, we can define α 1 (z) ( (1 ) 1 := z I Φ (Φ I) + zi) A 1 B α n (z) β 1,1 ( ) β 1,n ( ) 0 := e A I 0 0 β n,n ( ) Then (III5) holds with α (1) (z) L(z) := D 0 (z) 0 0 α (2) (z) α (n) (z) f( ) := [ β (1) ( ) β (2) ( ) β (n) ( ) ], where α (k) (z) := [ α k (z) α n (z) ] and β (k) ( ) := [ β k,k ( ) β k,n ( ) ] for k = 1,, n Proof: Since D is a scalar-valued function, it follows from [9, Theorem 432] that z 0 D is a zero of D if and only if 1/z 0 is an eigenvalue of p As D for all (, ), we can therefore choose an RH function whose zeros are the reciprocals of the unstable eigenvalues of p Moreover, since D = D 0, it follows that N N 0 = D 0 (P P 0 ) The realization ( F, Ḡ, H ) in (II4) shows that P (z) P 0 (z) equals ( (1 ) 1 (e A I) z I p (p I) + zi) A 1 B Thus a simple calculation gives (III5) IV EXACT BOUND FOR SCALAR SYSTEMS The objective in this section is to reduce the criterion (III3) to a computationally verifiable one for the following scalar plant: ẋ = ax + bu (a > 0) The extended system (II2) is given by ξ k+1 = p ξ k + b [ ] p u a p(1 + ) 1 k y k = [ 0 1 ] ξ k, (IV1) where p := e ah and := e a 1 In what follows we take b/a = 1 for simplicity of notation, because simultaneous stabilizability does not depend on this ratio Taking the Z-transform of (IV1) and then mapping z 1/z, we obtain the transfer function P : P (z) = (p 1)z 1 pz 1 pz (IV2) The extended system (IV1) is stabilizable and detectable except for = 1, at which point the system loses detectability From Assumption 21, we have (e ah 1, e ah 1) =: S max (IV3) The following theorem gives the exact bound on clock offset for scalar systems: Theorem 41: Let M 1 < 0 < M 2 and define S := [M 1, M 2 ] S max There exists a stabilizer of P for all S if and only if (p 1) 2 M 2 (p + 1) 2 M 1 < 4p (IV4) In particular, if M 1 = M 2 = M, then (IV4) is equivalent to M < 2p p (IV5) Changing the offset variable from = e A 1 to, we derive from (IV4) the maximum length of the offset interval [, ] allowed by using a linear time-invariant controller Corollary 42: Let < 0 < There exist a stabilizer of the extended system (II2) for all [, ] if and only if < 2 (log(p + 1) log(p 1)) /a We prove Theorem 41 by reducing the simultaneous stabilization problem to the Nevanlinna-Pick interpolation problem To this end, first we show that the stabilization problem is equivalent to an interpolation problem with a specified codomain: Lemma 43: Let M 1 < 0 < M 2 and let S := [M 1, M 2 ] S max There exists a stabilizer of P for S if and only if there exists F RH such that F is a map from D to C \ {(, 1/M 1 ] [1/M 2, )} and satisfies the interpolation conditions F (0) = 0, F (1) = 0, and F (1/p) = 1 Proof: We obtain an RH coprime factorization P = N /D with N := (p 1)z z α z α, D := 1 pz z α, (IV6) where α is a fixed complex number with α > 1 If we define X 0 and Y 0 by X 0 := 1 αp p 1, Y 0 := α, (IV7) then the Bezout identity N 0 X 0 + D 0 Y 0 = 1 holds Hence U and V in (III2) are given by U := 1 p 1 αp z(z 1) p 1 z α, V := p z(z 1)(1 pz) (z α) 2 Defining T 1 := p 1 αp z(z 1) p 1 z α, we therefore obtain T z(z 1)(1 pz) 2 := p (z α) 2, (IV8) U + QV = 1 (T 1 + QT 2 ) (Q RH ) (IV9) Define S + := S \ {0} Theorem 31 and (IV9) show that the plant P is simultaneously stabilizable by a single stabilizer if and only if there exists Q RH such that (1 (T 1 + QT 2 )) 1 RH ( S + ) (IV10) We have (IV10) if and only if 1 (T 1 + QT 2 ) has no zero in D for all S +, that is, T 1 (z) + Q(z)T 2 (z) C \ {(, 1/M 1 ] [1/M 2, )}

5 for all z D Now we prove the equivalence between the stability of Q and the interpolation conditions on F Suppose that Q RH and define F by F := T 1 + QT 2 RH Since the unstable zeros of T 2 are 0, 1, and 1/p and since Q is stable, it follows that F (0) = T 1 (0) = 0, F (1) = T 1 (1) = 0, and F (1/p) = T 1 (1/p) = 1 Conversely, let F RH satisfy F (0) = 0, F (1) = 0, and F (1/p) = 1 If we define Q := F T 1 T 2, (IV11) then Q belongs to RH In fact, assume that Q RH Since QT 2 = F T 1 RH, (IV12) it follows that Q has some unstable poles that are zeros of T 2 in D Let p 0 be one of the poles Since T 2 has only simple zeros in D, it follows that (QT 2 )(p 0 ) 0 The interpolation conditions of F lead to F (p 0 ) T 1 (p 0 ) = 0, which contradicts (IV12) This completes the proof Let us next solve the interpolation problem in Lemma 43 via the Nevanlinna-Pick interpolation We need to a conformal map from G := C \ {(, 1/M 1 ] [1/M 2, )} to D In [19], [20, Section 41], such a conformal map is given by 1 ϕ : G D : s 1 + ( ) 1/2 1 M2 s 1 M 1 s ( 1 M2 s 1 M 1 s ) 1/2 So far we consider finite-dimensional stabilizers, but in order to use the conformal map ϕ, we expand the class of stabilizers from RF to the field of fraction of the disk algebra That is, we study the following interpolation problem: Problem 44: Let z 1,, z n be distinct points in D and let w 1,, w n belong to G Find a function F such that F : D G, F is holomorphic in D and continuous in D, and F (z i ) = w i (i = 1,, n) Lemma 45: Problem 44 is equivalent to the problem of finding a function G such that G : D D, G is holomorphic in D and continuous in D, and G(z i ) = ϕ(w i ) (i = 1,, n) Proof: This follows from the fact that G = ϕ F Finally we obtain the proof of Theorem 41 Proof of Theorem 41: Lemmas 43 and 45 show that the stabilization problem with clock offset can be reduced to the Nevanlinna-Pick interpolation problem with a boundary condition [20, Chapter 2] We therefore obtain a necessary and sufficient condition based on the positive definiteness of the associated Pick matrix, which is equivalent to (IV4) The detailed calculation is omitted due to space constraints V COMPARISON IN THE CASE OF SCALAR PLANTS In this section, the bounds on the clock offset that can be allowed using a linear time-invariant stabilizer (obtained using Theorems 41 and 34) are compared with bounds that would be allowed by a static and a 2-periodic static state feedback stabilizer We also show a bound obtained using robust control by regarding the clock offset as an additive uncertainty The proposition below gives the exact bound on the clock offset that could be obtained using a static and a 2-periodic static stabilizer for a scalar plant Proposition 51: Define p := e ah, := e A 1, and S max as in (IV3) Consider the extended system P in (IV2) There exists a static stabilizer u k = Ky k of P for every S if and only if S satisfies S ( 1 p, 1 p ) (V1) Furthermore, fix S = [ M, M] S max for M > 0 There exists a 2-periodic static stabilizer uk K1 0 yk = (V2) u k+1 0 K 2 y k+1 of P for every S if and only if M < 1/(pL), where p2 + 1 p L := ( 2 1)p p 2 1 Proof: The proof is based on the Jury stability criterion, but is omitted due to space constraints A Comparison with static stabilizers and 2-periodic static stabilizers For all p > 1, a routine calculation shows that 1 p < 1 pl < 2p p (V3) As expected, (V1) results in the smallest range for values of because the set of all static stabilizers is a subset of the class of linear time-invariant stabilizers and that of 2-periodic static stabilizers in (V2) On the other hand, 2-periodic static stabilizers do not belong to the class of linear time-invariant stabilizers, but interestingly, the second inequality in (V3) always holds for all p > 1 B Sufficient condition from Theorem 34 From (IV6), we see that D = D 0 and N (z) N 0 (z) = z α Theorem 34 shows that there exists a stabilizer of P for all with < 1/p The bound is the same as (V1) by a static stabilizer This confirms that the use of the small gain theorem instead of the conformal map ϕ leads to a conservative stabilization analysis

6 - 044 Limit of from Assump$on Sta$c stabilizer Theorem 34 Robust stabiliza$on 2- periodic sta$c stabilizer Linear $me- invariant stabilizer Fig 3: Comparison among the bounds on obtained using each class of stabilizers C Sufficient condition from robust control of plants with additive uncertainties Taking P 0 as a nominal plant, an arbitrary plant P could be viewed as a perturbation of P 0 by the following additive uncertainty clock: P (z) P 0 (z) = = m(z)w (z), 1 pz where m(z) := z(p z)/(1 pz) and W (z) := p(z 1)/(p z) Since m(e jω ) = 1, it follows that P (e jω ) P 0 (e jω ) Wˆ(e jω ) for all e jω T and [ ˆ, ˆ] Hence there exists a stabilizer of P for [ ˆ, ˆ] if ˆ satisfies Wˆ (X 0 + QD 0 ) < 1 (V4) for some Q RH, where X 0 and D 0 are defined as in (IV7) and (IV6) Reducing (V4) to the Nevanlina-Pick interpolation problem as in [21], we see that (V4) is equivalent to ˆ < 1/p The bound derived from the conventional approach of robust control is also the same as (V1) by a static stabilizer Since in conventional robust control, we consider unnecessarily large class of uncertainties, it is not surprising to discover that this approach becomes conservative D Numerical example Set a = h = 1 The bound obtained using a linear timeinvariant stabilizer is < < and hence < < 1 The necessary condition in Proposition 33 does not give any bound on in this example Fig 3 shows the bounds on that can be allowed using each class of stabilizers We see that the bound obtained using a time-invariant stabilizer is smaller than the limit of from Assumption 21 It may be possible that periodic dynamic stabilizers achieve better performance as in [22] However, numerical computation shows that there does not exist a 2-periodic static stabilizer uk K1 0 yk = (V5) u k+1 K 3 K 2 y k+1 that stabilizes P for < < 1 (note that K 3 in (V5) may be nonzero) VI CONCLUDING REMARKS We studied the problem of stabilizing networked control systems with time-invariant clock offset We formulated the 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