On the design of compensator for quantization-caused ... - SpringerLink
SCIENCE CHINA Information Sciences
. RESEARCH PAPERS .
April 2011 Vol. 54 No. 4: 824–835 doi: 10.1007/s11432-010-4176-5
On the design of compensator for quantization-caused input-output deviation GUO YuQian∗ , GUI WeiHua & YANG ChunHua School of Information Science and Engineering, Central South University, Changsha 410083, China Received April 24, 2009; accepted October 12, 2009; published online February 28, 2011
Abstract This paper considers the design of compensators for systems with quantized inputs in order to reduce the influence of quantization. For systems with (vector) relative degrees, we propose a kind of compensators which can compensate for the accumulated output deviation completely caused by quantization. The proposed compensators are capable of keeping the differences of the input-output responses between the systems with quantized inputs and the original systems without considering quantization within certain small bounds. Simulations show that the compensators in this paper are robust with respect to model uncertainties, disturbance and measurement noise and can significantly improve the input-output responses of systems with both input quantization and packet dropouts. Keywords
networked control systems, quantization, packet dropout, I/O deviation compensation
Citation Guo Y Q, Gui W H, Yang C H. On the design of compensator for quantization-caused input-output deviation. Sci China Inf Sci, 2011, 54: 824–835, doi: 10.1007/s11432-010-4176-5
1
Introduction
Networked control systems (NCSs) have gained increasing attention in recent years. In NCSs, data are transmitted through digital channels and over network. As a result, the influence of information loss becomes a big concern in the design of NCSs. Information loss can be caused by many reasons including quantization, packet dropouts and time-delays, etc. These factors are basically related to the limited bandwidth of digital channels and reliability of network, etc. Quantization is a necessary procedure for analog signals to be transmitted over digital channels. Many quantized control problems are studied in literature, such as the coarsest static quantizer for stabilization [1–3], limited bandwidth [4–8], dynamic quantizer design [9] and robust control design [10, 11], etc. See also [12–21] for the recent development of networked control systems and quantized control systems. Because of the quantization error, there would be an output deviation between the system with quantized input and the original system without considering quantization. The output deviation may accumulate over time and cause large difference. Consequently, the controller designed for the original system without quantization could not be applied directly in general. One of the possible solutions is to apply compensators to reduce the influence of the quantization such that the input-output response of the ∗ Corresponding
author (email: [email protected])
c Science China Press and Springer-Verlag Berlin Heidelberg 2011
info.scichina.com
www.springerlink.com
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A system with quantized input and deviation compensator.
compensated system is as close as possible to that of the original system (see Figure 1). This is the main motivation of this paper. In this paper, we propose a procedure of compensator design which includes two steps: First, a duplicate of the original system is embedded into the compensator design in order to generate the expected output response. Then the output function of the compensator is designed to compensate for the accumulated part of the deviation completely. By doing this, the input-output response of the compensated system can be kept within a small neighborhood of that of the original system. We shall see that the radius of this neighborhood is related to the system matrices, relative degree and the coarseness of the quantizer. The study in this paper is related to the work in [9] where an optimal dynamic quantizer was reconstructed from a static one in order to reduce the influence of the latter. Actually, the dynamic quantizer in [9] is equivalent to a static quantizer plus a compensator. In our paper, different compensators are constructed from another perspective and both the systems with scalar relative degrees and vector relative degrees are considered. The main advantages of the compensators proposed in this paper are their robustness with respect to packet dropouts. We shall show this by simulation in the paper. This paper is organized as follows. In section 2, we set up the problem considered in this paper. In section 3 and section 4, compensators design for SISO and MIMO systems are considered respectively. In section 5, we discuss the systems with both input and measurement quantization. Illustrative examples are given in section 6. Section 7 is the concluding remarks. Notations. For any vector v = (v1 , . . . , vn )T and matrix M = (mij )n×m , v := maxi |vi | and M := maxi,j |mij |.
2
Problem setting
Consider the system
xk+1 = Axk + Buk , yk = C T xk + DT uk ,
(1)
with xk ∈ Rn , uk ∈ Rm , yk ∈ Rs and A, B, C and D constant matrices with compatible dimensions. It is assumed that m s, i.e., there are more outputs than inputs. Suppose that C = (c1 · · · cs ), D = (d1 · · · ds ). The counterpart of system (1) with quantized input is xk+1 = Axk + Bq(vk ), Σp : (2) yk = C T xk + DT q(vk ), where q(·) is a static quantizer with q(v) − v h. In [9], a kind of dynamic quantizer was proposed to keep the input-output response of the quantized system as close as possible to that of the system without considering quantization (see Figure 2). The optimal quantizer proposed in [9] can be viewed as a static quantizer q(·) plus a compensator, ξk+1 = Aξk + B(q(vk ) − uk ), (3) Σc : ηk = −(CB)−1 CAξk , with vk = uk + ηk . Note that this compensator is driven by the error q(vk ) − uk . When the input channel suffers from packet dropouts, the compensator of this structure may fail to effectively compensate for
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Figure 2
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Structure of the optimal dynamic quantizer proposed in [9].
the influence of information loss. (We shall use an example to show this in section 6). In this case, if the output of the plant is measurable, we can construct a compensator which is independent of the exact input of the plant. Precisely, the aim of this paper is to design a compensator (see Figure 1), ξk+1 = f (ξk , uk , yk ), (4) Σc : ηk = h(ξk , uk , yk ), such that the input-output response of systems (2)–(4) with vk = uk + ηk from uk to yk is as close as possible to the response of system (1) without considering quantization. For any sequence {uk }, denoted by the corresponding boldface letter uT the vector (u0 , u1 , . . . , uT ). We use yq (k, uk , x0 ) and y(k, uk , x0 ) to represent the output responses of the systems with and without quantization with respect to the same input {uk } and initial condition x0 respectively.
3
Single-input single-output systems
In this section, we investigate SISO systems with zero relative degrees first in subsection 3.1. Then in subsection 3.2, we consider more general cases. For SISO case, we write b, c and d instead of B, C and D respectively. 3.1
SISO systems with zero relative degrees
Since the plant has zero relative degree, we have d = 0. First of all, we assume that the state xk of the plant is available for feedback. In this case, we construct a compensator as follows: ⎧ ⎨ ξk+1 = Aξk + buk , (5) ⎩ ηk = 1 cT (ξk − xk ), ξ0 = x0 . d Theorem 1.
Under the compensator (5), there holds that, for any input {uk }, sup yq (k, uk , x0 ) − y(k, uk , x0 ) |d|h. k
Proof. Note that the first equation of compensator (5) is simply a duplicate of system (1) without considering quantization, it follows that y(k, uk , x0 ) = cT ξk + duk . Thus by the relation uk = vk − ηk , there holds yq (k, uk , x0 ) − y(k, uk , x0 ) = cT xk + dq(vk ) − (cT ξk + duk ) = d(q(vk ) − vk ). It follows that yq (k, uk , x0 ) − y(k, uk , x0 ) |d|h. Since the above inequality holds for any k and any input, it can be concluded that sup yq (k, uk , x0 ) − y(k, uk , x0 ) |d|h. k
(6)
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A system with deviation compensator and observer.
Remark 1. The compensator constructed above can be intuitively explained as follows: First, a duplicate of the plant is embedded into the compensator. Then formula (6) gives that yq (k, uk , x0 ) − y(k, uk , x0 ) = d(q(vk ) − vk ) + dηk − cT (ξk − xk ).
(7)
In above formula, the output deviation yq − y is divided into two parts: the first part d(q(vk ) − vk ) is the output deviation generated in the single step t = k, however, the second part dηk − cT (ξk − xk ) is the output deviation accumulated from t = 0 to t = k. Note that the first part cannot be compensated completely for arbitrary input [9]. Thus the output ηk of the compensator is just chosen to minimize the accumulated deviation. Since d = 0, this part of the deviation can be completely compensated for. Actually, it has been proved in [9] that the bound |d|h of the output deviation is optimal for all dynamic quantizers. When the state of the plant is not available for feedback, we have the following. Suppose that system (1) is observable and Σo is any observer such that
Theorem 2.
˜k ) = 0, lim (xk − x
k→∞
where x ˜k is the output of the observer (see Figure 3). Then under the compensator ⎧ ⎪ ⎨ ξk+1 = Aξk + buk , (8)
⎪ ⎩ ηk = 1 cT (ξk − x ˜k ), d there holds lim sup yq (k, uk , x0 ) − y(k, uk , ξ0 ) |d|h. k→∞
Proof.
Similarly, we have yq (k, uk , x0 ) − y(k, uk , ξ0 ) = d(q(vk ) − vk ) + cT (xk − x ˜k ) + dηk − cT (ξk − x ˜k ) ˜k ). = d(q(vk ) − vk ) + cT (xk − x
It follows that ˜k ) yq (k, uk , x0 ) − y(k, uk , ξ0 ) d(q(vk ) − vk ) + cT (xk − x ˜k ). |d|h + cT (xk − x ˜k ) tends to zero, we have Since the second part cT (xk − x lim sup yq (k, uk , x0 ) − y(k, uk , ξ0 ) |d|h. k→∞
3.2
SISO system with relative degree greater than 0
Suppose that system (1) has a relative degree ν 1. We construct a compensator as ⎧ ⎪ ⎨ ξk+1 = Aξk + buk , ⎪ ⎩ ηk =
1 cT Aν b
cT Aν (ξk − xk ),
(9) ξ0 = x0 .
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Theorem 3.
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Under the compensator (9), there holds that, for any input {uk }, sup yq (k, uk , x0 ) − y(k, uk , x0 ) |cT Aν−1 b|h. k
Proof.
Similarly, we have yq (k + ν, uk , x0 ) = cT Aν xk + cT Aν−1 bq(vk ), y(k + ν, uk , x0 ) = cT Aν ξk + cT Aν−1 buk .
It follows that yq (k + ν, uk , x0 ) − y(k + ν, uk , x0 ) = cT Aν xk + cT Aν−1 bq(vk ) − (cT Aν ξk + cT Aν−1 buk ) = cT Aν−1 b(q(vk ) − vk ) + cT Aν−1 bηk − cT Aν (ξk − xk ) = cT Aν−1 b(q(vk ) − vk ).
(10)
This implies that sup yq (k, uk , x0 ) − y(k, uk , x0 ) |cT Aν−1 b|h.
kν
When k < ν, there holds that yq (k, rk , x0 ) = cT Ak x0 ,
y(k, rk , x0 ) = cT Ak ξ0 .
Since ξ0 = x0 , we get yq (k, uk , x0 ) − y(k, uk , x0 ) = 0,
k = 0, . . . , ν − 1.
Thus sup yq (k, uk , x0 ) − y(k, uk , x0 ) |cT Aν−1 b|h. k
Remark 2. In fact, the output function of the compensator (9) is chosen to minimize the accumulated deviation cT Aν−1 bηk − cT Aν (ξk − xk ). The following result can be proved similarly. Theorem 4.
Suppose that system (1) is observable and Σo is any observer such that ˜k ) = 0, lim (xk − x
k→∞
where x ˜k is the output of the observer. Then under the compensator ⎧ ⎪ ⎨ ξk+1 = Aξk + buk , ⎪ ⎩ ηk =
1 cT Aν (ξk − x ˜k ), T c Aν−1 b
(11) ξ0 = x0 ,
there holds lim sup yq (k, uk , x0 ) − y(k, uk , ξ0 ) |cT Aν−1 b|h. k→∞
Proof.
Similarly, we have yq (k + ν, uk , x0 ) − y(k + ν, uk , x0 ) = cT Aν xk + cT Aν−1 bq(vk ) − (cT Aν ξk + cT Aν−1 buk ) ˜k ) = cT Aν−1 b(q(vk ) − vk ) + cT Aν (xk − x + cT Aν−1 bηk − cT Aν (ξk − x ˜k ) ˜k ). = cT Aν−1 b(q(vk ) − vk ) + cT Aν (xk − x
As a result, ˜k ) yq (k, uk , x0 ) − y(k, uk , ξ0 ) cT Aν−1 b(q(vk ) − vk ) + cT Aν (xk − x ˜k ). |cT Aν−1 b|h + cT Aν (xk − x ˜k ) tends to zero, there holds Since the second part cT Aν (xk − x lim sup yq (k, uk , x0 ) − y(k, uk , ξ0 ) |cT Aν−1 b|h. k→∞
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Multi-input multi-output systems with vector relative degree
Denote T , y1 y2 · · · ys T yq = yq1 yq2 · · · yqs . T Γ = (Aν1 )T c1 · · · (Aνs )T cs .
y=
Suppose that system (1) has a vector relative degree ν = (ν1 , . . . , νs ). In this case, V is of full column rank where T V = μ1 μ2 · · · μs (12)
and μT i We construct a compensator as
=
dT i ,
if νi = 0,
νi −1 B, cT i A
if νi 1.
ξk+1 = Aξk + Buk , ηk = [V T V ]−1 V T Γ(ξk − xk ),
(13)
ξ0 = x0 .
We explain this compensator as follows. First of all, the first equation of this compensator is still the duplicate of system (1). Secondly, it is easy to check that yq (k) = Γxk + V q(vk ), where
⎛
yq1 (k + ν1 , uk , x0 )
y(k) = Γξk + V uk . ⎛
⎞
⎟ ⎜ 2 ⎜ yq (k + ν2 , uk , x0 ) ⎟ ⎟ ⎜ yq (k) = ⎜ ⎟, .. ⎟ ⎜ . ⎠ ⎝ s yq (k + νs , uk , x0 )
y 1 (k + ν1 , uk , x0 )
⎜ 2 ⎜ y (k + ν2 , uk , x0 ) ⎜ y(k) = ⎜ .. ⎜ . ⎝ s y (k + νs , uk , x0 )
⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠
Thus yq (k) − y(k) = V (q(vk ) − vk ) + V ηk − Γ(ξk − xk ).
(14)
In order to compensate for the accumulated output error V ηk − Γ(ξk − xk ) as completely as possible, we choose ηk = arg min |V η − Γ(ξk − xk )|. η
By the relative degree condition, V is of full rank and we have ηk = [V T V ]−1 V T Γ(ξk − xk ). This coincides with the output function of the compensator (13). In the case of m = s, V is a square nonsingular matrix, the compensator (13) becomes ξk+1 = Aξk + Buk , ηk = V −1 Γ(ξk − xk ),
ξ0 = x0 .
(15)
We have the following result. Theorem 5. Suppose that system (1) has a vector relative degree ν = (ν1 , . . . , νs ) = 0 and m = s. Then under the compensator (24), there holds sup yq (k, uk , x0 ) − y(k, uk , x0 ) V h. k
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Proof.
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It is easy to check that V ηk − Γ(ξk − xk ) = 0,
thus by (14), yq (k) − y(k) = V (q(vk ) − vk ). So, yq (k) − y(k) V h, which is equivalent to yqi (k, uk , x0 ) − y i (k, uk , x0 ) V h,
k νi .
When 0 k < νi , there hold k yqi (k, uk , x0 ) = cT i A x0 ,
k y i (k, uk , x0 ) = cT i A ξ0 .
Since ξ0 = x0 , it follows that yqi (k, uk , x0 ) − y i (k, uk , x0 ) = 0,
k < νi .
Consequently, yqi (k, uk , x0 ) − y i (k, uk , x0 ) V h,
k 0,
or equivalently, sup yq (k, uk , x0 ) − y(k, uk , x0 ) V h. k
Similar to Theorem 2 and Theorem 4, we have Theorem 6. Assume that system (1) has a vector relative degree ν and is observable. Suppose that Σo is any observer such that ˜k ) = 0, lim (xk − x k→∞
where x ˜k is the output of the observer. In addition, assume that m = s. Then under the compensator ξk+1 = Aξk + Buk , (16) ˜k ), ηk = V −1 Γ(ξk − x there holds lim sup yq (k, uk , x0 ) − y(k, uk , ξ0 ) V h. k→∞
Proof.
Just follow the same arguement of Theorem 2 and Theorem 4.
Remark 3. to
If system (1) has zero relative degree, i.e., D = 0 is of full rank, the compensator degenerates
ξk+1 = Aξk + Buk , ηk = (DDT )−1 DC T (ξk − xk ).
(17)
If m = s, i.e., D is a square matrix, we have sup yq (k, uk , x0 ) − yq (k, uk , x0 ) Dh. k
(18)
Remark 4. In [9], the compensator is driven only by the quantization error q(v) − v. Thus one does not need to measure the output of the system. This is one of the merit of this structure especially when the output of the system is not measurable. In this paper, the compensator is driven by the state of the system, thus we require that the state of the system is either available directly or observable from the output. However, we can see from the construction of the compensator that, when system suffers from packet dropouts, our compensator is still capable of compensating for the accumulated output deviation whenever data are successfully transmitted.
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Systems with both input and measurement quantization
Consider systems with both input and measurement quantization (see Figure 4), the compensators constructed in this paper are still valid and capable of keeping the output deviation within a bound for arbitrary input. We only consider the SISO system with zero relative degree for simplicity. The other cases can be studied similarly. For a SISO system, the compensator constructed in Theorem 1 becomes ⎧ ⎨ ξk+1 = Aξk + buk , (19) ⎩ ηk = 1 cT [ξk − qo (xk )], d where qo (·) is the quantizer of the measurement xk satisfying qo (x) − x ho . In this case, y(k, uk , ξ0 ) = cT ξk + duk , yq (k, uk , x0 ) = cT xk + dq(vk ). Thus yq (k, uk , x0 ) − y(k, uk , ξ0 ) = d(q(vk ) − vk ) + cT (xk − qo (xk )). It follows that |yq (k, uk , x0 ) − y(k, uk , ξ0 )| |d|h + |c|ho holds for any input {uk }. Similarly, for SISO systems with relative degree ν > 0, the compensator becomes ⎧ ⎨ ξk+1 = Aξk + buk , 1 ⎩ ηk = cT Aν (ξk − qo (xk )). cT Aν b
(20)
In this case, there holds |yq (k, uk , x0 ) − y(k, uk , ξ0 )| |cT Aν−1 b|h + |cT Aν |ho . For MIMO systems with vector relative degree, the compensator becomes ξk+1 = Aξk + Buk , ηk = [V T V ]−1 V T Γ(ξk − qo (xk )),
(21)
and there holds yq (k, uk , x0 ) − y(k, uk , ξ0 ) V h + Γho .
6
(22)
Examples
Example 1. Consider a MIMO system depicted in Figure 5. Σp is the nominal system which is given by xk+1 = Axk + B1 v˜k + B2 wk , (23) yk = C T xk , where wk is the bounded disturbance which is ⎛ ⎛ ⎞ 0 0.95 1 0 ⎜ ⎜ ⎟ ⎜ A=⎜ −0.9 2 ⎟ ⎝ 0 ⎠ , B1 = ⎝ 0 1 0 0 0.9
assumed to satisfy |wk | 0.2 and ⎞ ⎛ ⎞ 0 1 ⎟ ⎜ ⎟ 0.01 0 0 T ⎜ ⎟ . 1⎟ ⎠ , B2 = ⎝ 0 ⎠ , C = 0 0 0.01 0 0
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Figure 4
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A system with both input and measurement
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April 2011 Vol. 54 No. 4
Figure 5
A MIMO system with uncertainty and
disturbance.
Σu (z) is the high frequency mode which is given by Σu (z) =
z −2 −1.2×10−4 z −1 +7×10−5 z −2 −5×10−5 z −1 +2×10−9
0
.
z −2 −1×10−4 z −1 +9×10−5 z −2 −4×10−5 z −1 +4×10−9
0
Thus we have v˜ = Σu (z)q(v), where q(·) is a static quantizer with h = 0.1. In the following, the compensator is designed based on the nominal plant and the simulations are however performed on the system with uncertainty, disturbance and measurement noise. It is easy to check that the nominal system has a vector relative degree ν = (2, 1) and V =
0
0.01
0.01
0
,
Γ=
0.009 0.0005 0.02 0
0
0.09
.
By the analysis of this paper, the compensator can be constructed as
ξk+1 = Aξk + B1 uk , ˜k )), ηk = V −1 Γ(ξk − qo (xk + w
(24)
where qo (·) is the measurement quantizer satisfying qo (x) − x 0.1 := ho , w ˜k is the measurement noise. By (22), if there are no uncertainty, disturbance and noise, then for any input {uk }, there holds yq (k) − y(k) V h + Γho = 0.01. In order to check the validity of the compensator, the quantization levels are chosen to be evenly distributed in simulation. The disturbance wk is a random sequence uniformly distributed in the interval [−0.2, 0.2] and w ˜k is assumed to be the Gaussian white noise with zero mean value and a variance 0.01. We use the sinusoid function 0.15 sin(0.1t) u= 0.15 sin(0.13t) as the test signal and the sampling period is chosen to be Ts = 1. The results are given in Figures 6–8. In Figure 8, we can see that the output deviation with respect to the original system without quantization is reduced significantly by the compensator.
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The first component of the output responses
Figure 7
to sinusoid input.
Figure 8
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The second component of the output respon-
ses to sinusoid input.
The output deviation yq (k) − y(k) with
respect to the system without quantization.
Figure 9
Output responses with respect to the sinusoid
input. The solid line is for the original system without quantization and packet dropout; the dash-dot line is for the system with the optimal dynamic quantizer proposed in [9]; the dash line is for the system with the compensator proposed in this paper.
Example 2. This example is to show that the proposed compensator in this paper is robust with respect to packet dropouts. Consider the system
xk+1 = Axk + bq(vk ), yk = cT xk ,
(25)
0.2 0 with cT = (1 1) and A = ( 0.8 0 0.9 ), b = ( 1 ). The quantization levels of the static quantizer q(·) is uniformly distributed and h = 0.1. Obviously, the system has relative degree 1. Thus by Theorem 3, the compensator is constructed as ξk+1 = Aξk + buk , (26) ηk = λT (ξk − qo (xk )),
where qo (·) is the measurement quantizer with ho = 0.1, λT = (0.8 dynamic quantizer can be constructed as
ξk+1 = Aξk + b(q(vk ) − vk ), v˜k = q(uk − λT ξk ).
1.1). According to [9], an optimal
(27)
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We assume that there are random packet dropouts in both input and measurement channels. The packet dropouts are independent of the state and assumed to be i.i.d. For the input channel, we assume that whenever data are lost, the input to the system becomes zero. For measurement channel, we assume that the output of the channel holds on it previous value whenever data are lost. It is also assumed that the sender sides do not know whether the data are received or not at each step. The probabilities of packet dropout for input and measurement channels are respectively 0.4 and 0.1. Simulation result about the responses of the system with respect to the sinusoid input is given in Figure 9. From the simulation, we can see that the optimal dynamic quantizer proposed in [9] fails to effectively reduce the output deviation caused by packet dropouts, however, the compensators proposed in this paper can compensate for the output deviation very effectively.
7
Concluding remarks
In this paper, we studied how to design compensators to reduce the influence of input and measurement information loss. For systems with (vector) relative degrees, we proposed a kind of compensators which can compensate for the accumulated output deviation completely. As a result, the compensated systems have similar input-output responses as the original systems without quantization. Especially, simulation results showed that the compensators proposed in this paper exhibit certain robustness with respect to model uncertainties, disturbance, measurement noise and packet dropouts.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 61074002, 61074001), and the Fundamental Research Funds for the Central Universities, China (Grant No. 2010QZZD016).
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15 Ishii H, Francis B A. Quadratic stabilization of sampled-data systems with quantization. Automatica, 2003, 39: 1793– 1800 16 Fagnani F, Zampieri S. Quantized stabilization of linear systems: complexity versus performance. IEEE Trans Autom Control, 2004, 49: 1534–1548 17 Qiang L, Lemmon M D. Stability of quantized control systems under dynamic bit assignment. IEEE Trans Autom Control, 2005, 50: 734–740 18 Bullo F, Liberzon D. Quantized control via locational optimization. IEEE Trans Autom Control, 2006, 51: 2–13 19 Delvenne J C. An optimal quantized feedback strategy for scalar linear systems. IEEE Trans Autom Control, 2006, 51: 298–303 20 Picasso B, Bicchi A. On the stabilization of linear systems under assigned I/O quantization. IEEE Trans Autom Control, 2007, 52: 1994–2000 21 Liberzon D. Quantization, time delays, and nonlinear stabilization. IEEE Trans Autom Control, 2006, 51: 1190–1195
. RESEARCH PAPERS .
April 2011 Vol. 54 No. 4: 824–835 doi: 10.1007/s11432-010-4176-5
On the design of compensator for quantization-caused input-output deviation GUO YuQian∗ , GUI WeiHua & YANG ChunHua School of Information Science and Engineering, Central South University, Changsha 410083, China Received April 24, 2009; accepted October 12, 2009; published online February 28, 2011
Abstract This paper considers the design of compensators for systems with quantized inputs in order to reduce the influence of quantization. For systems with (vector) relative degrees, we propose a kind of compensators which can compensate for the accumulated output deviation completely caused by quantization. The proposed compensators are capable of keeping the differences of the input-output responses between the systems with quantized inputs and the original systems without considering quantization within certain small bounds. Simulations show that the compensators in this paper are robust with respect to model uncertainties, disturbance and measurement noise and can significantly improve the input-output responses of systems with both input quantization and packet dropouts. Keywords
networked control systems, quantization, packet dropout, I/O deviation compensation
Citation Guo Y Q, Gui W H, Yang C H. On the design of compensator for quantization-caused input-output deviation. Sci China Inf Sci, 2011, 54: 824–835, doi: 10.1007/s11432-010-4176-5
1
Introduction
Networked control systems (NCSs) have gained increasing attention in recent years. In NCSs, data are transmitted through digital channels and over network. As a result, the influence of information loss becomes a big concern in the design of NCSs. Information loss can be caused by many reasons including quantization, packet dropouts and time-delays, etc. These factors are basically related to the limited bandwidth of digital channels and reliability of network, etc. Quantization is a necessary procedure for analog signals to be transmitted over digital channels. Many quantized control problems are studied in literature, such as the coarsest static quantizer for stabilization [1–3], limited bandwidth [4–8], dynamic quantizer design [9] and robust control design [10, 11], etc. See also [12–21] for the recent development of networked control systems and quantized control systems. Because of the quantization error, there would be an output deviation between the system with quantized input and the original system without considering quantization. The output deviation may accumulate over time and cause large difference. Consequently, the controller designed for the original system without quantization could not be applied directly in general. One of the possible solutions is to apply compensators to reduce the influence of the quantization such that the input-output response of the ∗ Corresponding
author (email: [email protected])
c Science China Press and Springer-Verlag Berlin Heidelberg 2011
info.scichina.com
www.springerlink.com
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A system with quantized input and deviation compensator.
compensated system is as close as possible to that of the original system (see Figure 1). This is the main motivation of this paper. In this paper, we propose a procedure of compensator design which includes two steps: First, a duplicate of the original system is embedded into the compensator design in order to generate the expected output response. Then the output function of the compensator is designed to compensate for the accumulated part of the deviation completely. By doing this, the input-output response of the compensated system can be kept within a small neighborhood of that of the original system. We shall see that the radius of this neighborhood is related to the system matrices, relative degree and the coarseness of the quantizer. The study in this paper is related to the work in [9] where an optimal dynamic quantizer was reconstructed from a static one in order to reduce the influence of the latter. Actually, the dynamic quantizer in [9] is equivalent to a static quantizer plus a compensator. In our paper, different compensators are constructed from another perspective and both the systems with scalar relative degrees and vector relative degrees are considered. The main advantages of the compensators proposed in this paper are their robustness with respect to packet dropouts. We shall show this by simulation in the paper. This paper is organized as follows. In section 2, we set up the problem considered in this paper. In section 3 and section 4, compensators design for SISO and MIMO systems are considered respectively. In section 5, we discuss the systems with both input and measurement quantization. Illustrative examples are given in section 6. Section 7 is the concluding remarks. Notations. For any vector v = (v1 , . . . , vn )T and matrix M = (mij )n×m , v := maxi |vi | and M := maxi,j |mij |.
2
Problem setting
Consider the system
xk+1 = Axk + Buk , yk = C T xk + DT uk ,
(1)
with xk ∈ Rn , uk ∈ Rm , yk ∈ Rs and A, B, C and D constant matrices with compatible dimensions. It is assumed that m s, i.e., there are more outputs than inputs. Suppose that C = (c1 · · · cs ), D = (d1 · · · ds ). The counterpart of system (1) with quantized input is xk+1 = Axk + Bq(vk ), Σp : (2) yk = C T xk + DT q(vk ), where q(·) is a static quantizer with q(v) − v h. In [9], a kind of dynamic quantizer was proposed to keep the input-output response of the quantized system as close as possible to that of the system without considering quantization (see Figure 2). The optimal quantizer proposed in [9] can be viewed as a static quantizer q(·) plus a compensator, ξk+1 = Aξk + B(q(vk ) − uk ), (3) Σc : ηk = −(CB)−1 CAξk , with vk = uk + ηk . Note that this compensator is driven by the error q(vk ) − uk . When the input channel suffers from packet dropouts, the compensator of this structure may fail to effectively compensate for
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Figure 2
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Structure of the optimal dynamic quantizer proposed in [9].
the influence of information loss. (We shall use an example to show this in section 6). In this case, if the output of the plant is measurable, we can construct a compensator which is independent of the exact input of the plant. Precisely, the aim of this paper is to design a compensator (see Figure 1), ξk+1 = f (ξk , uk , yk ), (4) Σc : ηk = h(ξk , uk , yk ), such that the input-output response of systems (2)–(4) with vk = uk + ηk from uk to yk is as close as possible to the response of system (1) without considering quantization. For any sequence {uk }, denoted by the corresponding boldface letter uT the vector (u0 , u1 , . . . , uT ). We use yq (k, uk , x0 ) and y(k, uk , x0 ) to represent the output responses of the systems with and without quantization with respect to the same input {uk } and initial condition x0 respectively.
3
Single-input single-output systems
In this section, we investigate SISO systems with zero relative degrees first in subsection 3.1. Then in subsection 3.2, we consider more general cases. For SISO case, we write b, c and d instead of B, C and D respectively. 3.1
SISO systems with zero relative degrees
Since the plant has zero relative degree, we have d = 0. First of all, we assume that the state xk of the plant is available for feedback. In this case, we construct a compensator as follows: ⎧ ⎨ ξk+1 = Aξk + buk , (5) ⎩ ηk = 1 cT (ξk − xk ), ξ0 = x0 . d Theorem 1.
Under the compensator (5), there holds that, for any input {uk }, sup yq (k, uk , x0 ) − y(k, uk , x0 ) |d|h. k
Proof. Note that the first equation of compensator (5) is simply a duplicate of system (1) without considering quantization, it follows that y(k, uk , x0 ) = cT ξk + duk . Thus by the relation uk = vk − ηk , there holds yq (k, uk , x0 ) − y(k, uk , x0 ) = cT xk + dq(vk ) − (cT ξk + duk ) = d(q(vk ) − vk ). It follows that yq (k, uk , x0 ) − y(k, uk , x0 ) |d|h. Since the above inequality holds for any k and any input, it can be concluded that sup yq (k, uk , x0 ) − y(k, uk , x0 ) |d|h. k
(6)
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A system with deviation compensator and observer.
Remark 1. The compensator constructed above can be intuitively explained as follows: First, a duplicate of the plant is embedded into the compensator. Then formula (6) gives that yq (k, uk , x0 ) − y(k, uk , x0 ) = d(q(vk ) − vk ) + dηk − cT (ξk − xk ).
(7)
In above formula, the output deviation yq − y is divided into two parts: the first part d(q(vk ) − vk ) is the output deviation generated in the single step t = k, however, the second part dηk − cT (ξk − xk ) is the output deviation accumulated from t = 0 to t = k. Note that the first part cannot be compensated completely for arbitrary input [9]. Thus the output ηk of the compensator is just chosen to minimize the accumulated deviation. Since d = 0, this part of the deviation can be completely compensated for. Actually, it has been proved in [9] that the bound |d|h of the output deviation is optimal for all dynamic quantizers. When the state of the plant is not available for feedback, we have the following. Suppose that system (1) is observable and Σo is any observer such that
Theorem 2.
˜k ) = 0, lim (xk − x
k→∞
where x ˜k is the output of the observer (see Figure 3). Then under the compensator ⎧ ⎪ ⎨ ξk+1 = Aξk + buk , (8)
⎪ ⎩ ηk = 1 cT (ξk − x ˜k ), d there holds lim sup yq (k, uk , x0 ) − y(k, uk , ξ0 ) |d|h. k→∞
Proof.
Similarly, we have yq (k, uk , x0 ) − y(k, uk , ξ0 ) = d(q(vk ) − vk ) + cT (xk − x ˜k ) + dηk − cT (ξk − x ˜k ) ˜k ). = d(q(vk ) − vk ) + cT (xk − x
It follows that ˜k ) yq (k, uk , x0 ) − y(k, uk , ξ0 ) d(q(vk ) − vk ) + cT (xk − x ˜k ). |d|h + cT (xk − x ˜k ) tends to zero, we have Since the second part cT (xk − x lim sup yq (k, uk , x0 ) − y(k, uk , ξ0 ) |d|h. k→∞
3.2
SISO system with relative degree greater than 0
Suppose that system (1) has a relative degree ν 1. We construct a compensator as ⎧ ⎪ ⎨ ξk+1 = Aξk + buk , ⎪ ⎩ ηk =
1 cT Aν b
cT Aν (ξk − xk ),
(9) ξ0 = x0 .
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Theorem 3.
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Under the compensator (9), there holds that, for any input {uk }, sup yq (k, uk , x0 ) − y(k, uk , x0 ) |cT Aν−1 b|h. k
Proof.
Similarly, we have yq (k + ν, uk , x0 ) = cT Aν xk + cT Aν−1 bq(vk ), y(k + ν, uk , x0 ) = cT Aν ξk + cT Aν−1 buk .
It follows that yq (k + ν, uk , x0 ) − y(k + ν, uk , x0 ) = cT Aν xk + cT Aν−1 bq(vk ) − (cT Aν ξk + cT Aν−1 buk ) = cT Aν−1 b(q(vk ) − vk ) + cT Aν−1 bηk − cT Aν (ξk − xk ) = cT Aν−1 b(q(vk ) − vk ).
(10)
This implies that sup yq (k, uk , x0 ) − y(k, uk , x0 ) |cT Aν−1 b|h.
kν
When k < ν, there holds that yq (k, rk , x0 ) = cT Ak x0 ,
y(k, rk , x0 ) = cT Ak ξ0 .
Since ξ0 = x0 , we get yq (k, uk , x0 ) − y(k, uk , x0 ) = 0,
k = 0, . . . , ν − 1.
Thus sup yq (k, uk , x0 ) − y(k, uk , x0 ) |cT Aν−1 b|h. k
Remark 2. In fact, the output function of the compensator (9) is chosen to minimize the accumulated deviation cT Aν−1 bηk − cT Aν (ξk − xk ). The following result can be proved similarly. Theorem 4.
Suppose that system (1) is observable and Σo is any observer such that ˜k ) = 0, lim (xk − x
k→∞
where x ˜k is the output of the observer. Then under the compensator ⎧ ⎪ ⎨ ξk+1 = Aξk + buk , ⎪ ⎩ ηk =
1 cT Aν (ξk − x ˜k ), T c Aν−1 b
(11) ξ0 = x0 ,
there holds lim sup yq (k, uk , x0 ) − y(k, uk , ξ0 ) |cT Aν−1 b|h. k→∞
Proof.
Similarly, we have yq (k + ν, uk , x0 ) − y(k + ν, uk , x0 ) = cT Aν xk + cT Aν−1 bq(vk ) − (cT Aν ξk + cT Aν−1 buk ) ˜k ) = cT Aν−1 b(q(vk ) − vk ) + cT Aν (xk − x + cT Aν−1 bηk − cT Aν (ξk − x ˜k ) ˜k ). = cT Aν−1 b(q(vk ) − vk ) + cT Aν (xk − x
As a result, ˜k ) yq (k, uk , x0 ) − y(k, uk , ξ0 ) cT Aν−1 b(q(vk ) − vk ) + cT Aν (xk − x ˜k ). |cT Aν−1 b|h + cT Aν (xk − x ˜k ) tends to zero, there holds Since the second part cT Aν (xk − x lim sup yq (k, uk , x0 ) − y(k, uk , ξ0 ) |cT Aν−1 b|h. k→∞
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Multi-input multi-output systems with vector relative degree
Denote T , y1 y2 · · · ys T yq = yq1 yq2 · · · yqs . T Γ = (Aν1 )T c1 · · · (Aνs )T cs .
y=
Suppose that system (1) has a vector relative degree ν = (ν1 , . . . , νs ). In this case, V is of full column rank where T V = μ1 μ2 · · · μs (12)
and μT i We construct a compensator as
=
dT i ,
if νi = 0,
νi −1 B, cT i A
if νi 1.
ξk+1 = Aξk + Buk , ηk = [V T V ]−1 V T Γ(ξk − xk ),
(13)
ξ0 = x0 .
We explain this compensator as follows. First of all, the first equation of this compensator is still the duplicate of system (1). Secondly, it is easy to check that yq (k) = Γxk + V q(vk ), where
⎛
yq1 (k + ν1 , uk , x0 )
y(k) = Γξk + V uk . ⎛
⎞
⎟ ⎜ 2 ⎜ yq (k + ν2 , uk , x0 ) ⎟ ⎟ ⎜ yq (k) = ⎜ ⎟, .. ⎟ ⎜ . ⎠ ⎝ s yq (k + νs , uk , x0 )
y 1 (k + ν1 , uk , x0 )
⎜ 2 ⎜ y (k + ν2 , uk , x0 ) ⎜ y(k) = ⎜ .. ⎜ . ⎝ s y (k + νs , uk , x0 )
⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠
Thus yq (k) − y(k) = V (q(vk ) − vk ) + V ηk − Γ(ξk − xk ).
(14)
In order to compensate for the accumulated output error V ηk − Γ(ξk − xk ) as completely as possible, we choose ηk = arg min |V η − Γ(ξk − xk )|. η
By the relative degree condition, V is of full rank and we have ηk = [V T V ]−1 V T Γ(ξk − xk ). This coincides with the output function of the compensator (13). In the case of m = s, V is a square nonsingular matrix, the compensator (13) becomes ξk+1 = Aξk + Buk , ηk = V −1 Γ(ξk − xk ),
ξ0 = x0 .
(15)
We have the following result. Theorem 5. Suppose that system (1) has a vector relative degree ν = (ν1 , . . . , νs ) = 0 and m = s. Then under the compensator (24), there holds sup yq (k, uk , x0 ) − y(k, uk , x0 ) V h. k
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Proof.
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It is easy to check that V ηk − Γ(ξk − xk ) = 0,
thus by (14), yq (k) − y(k) = V (q(vk ) − vk ). So, yq (k) − y(k) V h, which is equivalent to yqi (k, uk , x0 ) − y i (k, uk , x0 ) V h,
k νi .
When 0 k < νi , there hold k yqi (k, uk , x0 ) = cT i A x0 ,
k y i (k, uk , x0 ) = cT i A ξ0 .
Since ξ0 = x0 , it follows that yqi (k, uk , x0 ) − y i (k, uk , x0 ) = 0,
k < νi .
Consequently, yqi (k, uk , x0 ) − y i (k, uk , x0 ) V h,
k 0,
or equivalently, sup yq (k, uk , x0 ) − y(k, uk , x0 ) V h. k
Similar to Theorem 2 and Theorem 4, we have Theorem 6. Assume that system (1) has a vector relative degree ν and is observable. Suppose that Σo is any observer such that ˜k ) = 0, lim (xk − x k→∞
where x ˜k is the output of the observer. In addition, assume that m = s. Then under the compensator ξk+1 = Aξk + Buk , (16) ˜k ), ηk = V −1 Γ(ξk − x there holds lim sup yq (k, uk , x0 ) − y(k, uk , ξ0 ) V h. k→∞
Proof.
Just follow the same arguement of Theorem 2 and Theorem 4.
Remark 3. to
If system (1) has zero relative degree, i.e., D = 0 is of full rank, the compensator degenerates
ξk+1 = Aξk + Buk , ηk = (DDT )−1 DC T (ξk − xk ).
(17)
If m = s, i.e., D is a square matrix, we have sup yq (k, uk , x0 ) − yq (k, uk , x0 ) Dh. k
(18)
Remark 4. In [9], the compensator is driven only by the quantization error q(v) − v. Thus one does not need to measure the output of the system. This is one of the merit of this structure especially when the output of the system is not measurable. In this paper, the compensator is driven by the state of the system, thus we require that the state of the system is either available directly or observable from the output. However, we can see from the construction of the compensator that, when system suffers from packet dropouts, our compensator is still capable of compensating for the accumulated output deviation whenever data are successfully transmitted.
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Systems with both input and measurement quantization
Consider systems with both input and measurement quantization (see Figure 4), the compensators constructed in this paper are still valid and capable of keeping the output deviation within a bound for arbitrary input. We only consider the SISO system with zero relative degree for simplicity. The other cases can be studied similarly. For a SISO system, the compensator constructed in Theorem 1 becomes ⎧ ⎨ ξk+1 = Aξk + buk , (19) ⎩ ηk = 1 cT [ξk − qo (xk )], d where qo (·) is the quantizer of the measurement xk satisfying qo (x) − x ho . In this case, y(k, uk , ξ0 ) = cT ξk + duk , yq (k, uk , x0 ) = cT xk + dq(vk ). Thus yq (k, uk , x0 ) − y(k, uk , ξ0 ) = d(q(vk ) − vk ) + cT (xk − qo (xk )). It follows that |yq (k, uk , x0 ) − y(k, uk , ξ0 )| |d|h + |c|ho holds for any input {uk }. Similarly, for SISO systems with relative degree ν > 0, the compensator becomes ⎧ ⎨ ξk+1 = Aξk + buk , 1 ⎩ ηk = cT Aν (ξk − qo (xk )). cT Aν b
(20)
In this case, there holds |yq (k, uk , x0 ) − y(k, uk , ξ0 )| |cT Aν−1 b|h + |cT Aν |ho . For MIMO systems with vector relative degree, the compensator becomes ξk+1 = Aξk + Buk , ηk = [V T V ]−1 V T Γ(ξk − qo (xk )),
(21)
and there holds yq (k, uk , x0 ) − y(k, uk , ξ0 ) V h + Γho .
6
(22)
Examples
Example 1. Consider a MIMO system depicted in Figure 5. Σp is the nominal system which is given by xk+1 = Axk + B1 v˜k + B2 wk , (23) yk = C T xk , where wk is the bounded disturbance which is ⎛ ⎛ ⎞ 0 0.95 1 0 ⎜ ⎜ ⎟ ⎜ A=⎜ −0.9 2 ⎟ ⎝ 0 ⎠ , B1 = ⎝ 0 1 0 0 0.9
assumed to satisfy |wk | 0.2 and ⎞ ⎛ ⎞ 0 1 ⎟ ⎜ ⎟ 0.01 0 0 T ⎜ ⎟ . 1⎟ ⎠ , B2 = ⎝ 0 ⎠ , C = 0 0 0.01 0 0
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Figure 4
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A system with both input and measurement
quantization.
April 2011 Vol. 54 No. 4
Figure 5
A MIMO system with uncertainty and
disturbance.
Σu (z) is the high frequency mode which is given by Σu (z) =
z −2 −1.2×10−4 z −1 +7×10−5 z −2 −5×10−5 z −1 +2×10−9
0
.
z −2 −1×10−4 z −1 +9×10−5 z −2 −4×10−5 z −1 +4×10−9
0
Thus we have v˜ = Σu (z)q(v), where q(·) is a static quantizer with h = 0.1. In the following, the compensator is designed based on the nominal plant and the simulations are however performed on the system with uncertainty, disturbance and measurement noise. It is easy to check that the nominal system has a vector relative degree ν = (2, 1) and V =
0
0.01
0.01
0
,
Γ=
0.009 0.0005 0.02 0
0
0.09
.
By the analysis of this paper, the compensator can be constructed as
ξk+1 = Aξk + B1 uk , ˜k )), ηk = V −1 Γ(ξk − qo (xk + w
(24)
where qo (·) is the measurement quantizer satisfying qo (x) − x 0.1 := ho , w ˜k is the measurement noise. By (22), if there are no uncertainty, disturbance and noise, then for any input {uk }, there holds yq (k) − y(k) V h + Γho = 0.01. In order to check the validity of the compensator, the quantization levels are chosen to be evenly distributed in simulation. The disturbance wk is a random sequence uniformly distributed in the interval [−0.2, 0.2] and w ˜k is assumed to be the Gaussian white noise with zero mean value and a variance 0.01. We use the sinusoid function 0.15 sin(0.1t) u= 0.15 sin(0.13t) as the test signal and the sampling period is chosen to be Ts = 1. The results are given in Figures 6–8. In Figure 8, we can see that the output deviation with respect to the original system without quantization is reduced significantly by the compensator.
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The first component of the output responses
Figure 7
to sinusoid input.
Figure 8
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The second component of the output respon-
ses to sinusoid input.
The output deviation yq (k) − y(k) with
respect to the system without quantization.
Figure 9
Output responses with respect to the sinusoid
input. The solid line is for the original system without quantization and packet dropout; the dash-dot line is for the system with the optimal dynamic quantizer proposed in [9]; the dash line is for the system with the compensator proposed in this paper.
Example 2. This example is to show that the proposed compensator in this paper is robust with respect to packet dropouts. Consider the system
xk+1 = Axk + bq(vk ), yk = cT xk ,
(25)
0.2 0 with cT = (1 1) and A = ( 0.8 0 0.9 ), b = ( 1 ). The quantization levels of the static quantizer q(·) is uniformly distributed and h = 0.1. Obviously, the system has relative degree 1. Thus by Theorem 3, the compensator is constructed as ξk+1 = Aξk + buk , (26) ηk = λT (ξk − qo (xk )),
where qo (·) is the measurement quantizer with ho = 0.1, λT = (0.8 dynamic quantizer can be constructed as
ξk+1 = Aξk + b(q(vk ) − vk ), v˜k = q(uk − λT ξk ).
1.1). According to [9], an optimal
(27)
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We assume that there are random packet dropouts in both input and measurement channels. The packet dropouts are independent of the state and assumed to be i.i.d. For the input channel, we assume that whenever data are lost, the input to the system becomes zero. For measurement channel, we assume that the output of the channel holds on it previous value whenever data are lost. It is also assumed that the sender sides do not know whether the data are received or not at each step. The probabilities of packet dropout for input and measurement channels are respectively 0.4 and 0.1. Simulation result about the responses of the system with respect to the sinusoid input is given in Figure 9. From the simulation, we can see that the optimal dynamic quantizer proposed in [9] fails to effectively reduce the output deviation caused by packet dropouts, however, the compensators proposed in this paper can compensate for the output deviation very effectively.
7
Concluding remarks
In this paper, we studied how to design compensators to reduce the influence of input and measurement information loss. For systems with (vector) relative degrees, we proposed a kind of compensators which can compensate for the accumulated output deviation completely. As a result, the compensated systems have similar input-output responses as the original systems without quantization. Especially, simulation results showed that the compensators proposed in this paper exhibit certain robustness with respect to model uncertainties, disturbance, measurement noise and packet dropouts.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 61074002, 61074001), and the Fundamental Research Funds for the Central Universities, China (Grant No. 2010QZZD016).
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