Multiple descriptions for packetized predictive control
- Jan Østergaard^{1}Email author and
- Daniel Quevedo^{2}
DOI: 10.1186/s13634-016-0343-1
© Østergaard and Quevedo. 2016
Received: 21 October 2015
Accepted: 28 March 2016
Published: 11 April 2016
Abstract
In this paper, we propose to use multiple descriptions (MDs) to achieve a high degree of robustness towards random packet delays and erasures in networked control systems. In particular, we consider the scenario, where a data-rate limited channel is located between the controller and the plant input. This forward channel also introduces random delays and dropouts. The feedback channel from the plant output to the controller is assumed noiseless. We show how to design MDs for packetized predicted control (PPC) in order to enhance the robustness. In the proposed scheme, a quantized control vector with future tentative control signals is transmitted to the plant at each discrete time instant. This control vector is then transformed into M redundant descriptions (packets) such that when receiving any 1≤J≤M packets, the current control signal as well as J−1 future control signals can be reliably reconstructed at the plant side. For the particular case of LTI plant models and i.i.d. channels, we show that the overall system forms a Markov jump linear system. We provide conditions for mean square stability and derive upper bounds on the operational bit rate of the quantizer to guarantee a desired performance level. Simulations reveal that a significant gain over conventional PPC can be achieved when combining PPC with suitably designed MDs.
Keywords
Quantization Networked control Multiple descriptions1 Introduction
In networked control systems (NCSs), the controller communicates with the plant via a general purpose communication network [1, 2]. When compared to using dedicated hardwired control networks, the use of general purpose and possibly wireless communication technology brings significant benefits in terms of efficiency, interoperability, deployment costs, etc. However, the use of practical communication technology also leads to new challenges, since the network needs to be taken into account in the overall design, see also [1–7].
In this paper, we will focus on the existence of a digital network between the controller and the plant input. This network contains either a single channel that introduces i.i.d. packet delays and erasures or multiple independent channels with i.i.d. packet delays and erasures. The channel between the plant output and the controller is considered ideal, i.e., noiseless and instantaneous. For example, this could be a situation where the controller and plant communicates over wireless channels. The controller could be battery driven and therefore with limited transmission power. On the other hand, the plant might not have a limitation on the transmission power. In this case, the reverse channel from the plant to the controller has a significantly greater SNR than the forward channel between the controller and the plant. There are many other practical sitations with wireless controller-actuator links but direct sensor-controller connections, e.g., groups of agents/vehicles/robots/drones. Their positions/formation are sensed via a system comprising a camera and attached controller. Activation commands are then sent wirelessly to the agents.
The main contributions of this work is the theoretical analysis and practical design of the quantized control signals. In particular, we propose to combine a recent robust control strategy known as (quantized) packet predictive control (PPC) [8–11] with a joint source-channel coding strategy based on multiple description (MD) coding [12, 13]. We provide computable upper bounds on the operational bit rate required for coding the quantized control signals (descriptions) and provide a practical design based on our theoretical analysis. The simulation study shows that the combination of MDs and PPC provides a significant improvement over PPC in the case of large packet loss ratios.
In quantized PPC, a control vector with the current and N−1 future predicted plant inputs is constructed at the controller side to compensate for random delays and packet dropouts in the channel. Thus, in the case of packet erasures (and if not too many consecutive dropouts occur), the buffer will feed the plant with the appropriate future predicted control value [8]. The key principle of MDs is to encode a source signal into a number of descriptions (packets) that are transmitted over separate channels. Each description is able to approximate the source signal to within a prescribed quality. Moreover, if several descriptions are received, they can be combined to further improve the reconstruction quality. Thus, in the case of packet erasures, it is possible to achieve a graceful degradation of the reconstruction quality [13].
The design of optimal quantized control strategies subject to data rate limitations defines a complicated problem that lies in the intersection of signal processing and controls. In particular, if the quantizers are designed using conventional open-loop source-coding strategies, it cannot be guaranteed that the overall system will be stable, when used in closed-loop control. Indeed, the resulting data rate could exceed the bandwidth of the digital channel, the data rate could be too low to capture the plant uncertainty and thereby not guarantee stability, or the non-linear effects due to quantization could have a negative impact on the overall stability when fed back into the system [8, 9, 14, 15].
The combination of MDs and PPC has to the best of the authors’ knowledge not been considered before (except in the conference contributions of the authors [16–18]). In [16], MDs were used for power control in wireless sensor networks. The quantizers were designed under high-resolution assumptions, and no stability assessment was provided. In [17, 18], the preliminary ideas for the current work (without analysis and proofs) were presented. MDs for state-estimation was considered in [19, 20] under high-resolution quantization assumptions. The design of lattice quantizers for PPC without MDs was treated in [16, 21] for the cases of entropy-constrained and resolution-constrained quantization, respectively.
In this work, we will focus on LTI plant models, which are (possibly) open-loop unstable. Thus, it is necessary to provide quantized control signals to the plant in a reliable way to guarantee stability in the presence of data rate limitations, random packet delays and erasures. Our key idea is to design and use MDs in a novel way that differs from how it is traditionally used. Traditionally, when the received descriptions are combined at the decoder, the approximation of a given source signal is improved. On the other hand, in the proposed work, when the received descriptions are combined at the decoder, then rather than improving existing control signals, new future controls signals are instead recovered.
There exists a vast amount on literature on MJLS with delays, cf.,[22–25]. In the present work, we show that the overall system with delays, erasures, quantization effects, and multiple descriptions, can be cast as a Markov jump linear system (MJLS), which makes it possible to use general stability results from the MJLS literature [26, 27].
The paper is organized as follows. Section 2 contains background information on quantized PPC. Section 3 contains the system analysis of a theoretical joint PPC and MD scheme. Section 4 presents the design of the combined practical PPC and MD scheme. Section 5 provides a simulation study of the proposed scheme. Section 6 contains the conclusions. Proofs of lemmas and theorems are deferred to the appendices.
1.1 Notation
Let S ^{ ↓ } be the down-shift-by-one matrix operator, which replaces the jth row of an N×M matrix by its (j−1)th row for \(j=N,\dotsc, 2\). Similarly, define S ^{ ↑ } as the up-shift-by-one matrix operator. Let e _{ i } denote the unit-vector aligned with the ith axis of the Cartesian coordinate system, e.g., e _{2}=[0,1,0,⋯,0]^{ T }, where the dimension of e _{ i } will be clear from the context. Let \(\boldsymbol {1}_{i}\in \mathbb {R}^{i}\) be the all-ones vector of dimension i. Let γ _{ i } be the matrix operator that takes the ith diagonal of an N×N matrix, where i=1 is the main diagonal and i>1 are diagonals above the main diagonal. Thus, \(\gamma _{i}(A) \in \mathbb {R}^{N-i+1}\) if \(A \in \mathbb {R}^{N\times N}\). We will use σ _{ r }(A) to denote the spectral radius of the matrix A, and A⊗B denotes the usual Kronecker product between the matrices A and B. The squared and weighted l _{2}-norm of a vector, say x, is written as \(\|x \|_{P}^{2} = x^{T} P x\), where P≽0, i.e., P is a positive semidefinite matrix.
2 Quantized packetized control over erasure channels
2.1 System model
In (1), \(w_{t}\in \mathbb {R}^{z'}\), z ^{′}≥1, is an unmeasured disturbance, modeled as an arbitrarily distributed (and with possibly unbounded support) zero-mean stochastic process with bounded covariance matrix Σ _{ w }, and \(B_{1}\in \mathbb {R}^{z}\) and \(B_{2}\in \mathbb {R}^{z\times z'}\). We do not assume that \(A\in \mathbb {R}^{z\times z}\) is stable; however, we will assume that the pair (A,B _{1}) is stabilizable. The initial state x _{0} is arbitrarily distributed with bounded variance.
2.2 Cost function
which exists if the system (1) is stabilizable [28].
where p _{ ℓ } denotes the probability of using the control signal u ℓ′. Moreover, in this work, we will also model the effect of the quantizer directly in the design of the control signal u ℓ′, see Section 2.4 for details.
Following the ideas underlying PPCs, see, e.g., [29], at each time instant t, and for current state x _{ t }, the controller sends the entire optimizing sequence, \(\bar {u}_{t}\), to the actuator node. Depending upon future packet dropout scenarios, a subsequence of \(\bar {u}_{t}\) will be applied at the plant input, or not. Following the receding horizon paradigm, at the next time instant, x _{ t+1} is used to carry out another optimization, yielding \(\bar {u}_{t+1}\), etc.
2.3 Network effects
Finally, we assume that the channel statistics are stationary so that \(\text {Prob}\left (d^{i}_{t,t'} = 1 | d^{i}_{t,t'-1}, d^{i}_{t,t'-2}, \dotsc, d^{i}_{t,t' - N+1}\right)\) does not depend upon t. We will make explicit use of the above stationarity and Markov assumptions in Lemma 3.2.
2.4 Quantization constraints
We consider a bit-rate limited digital network between controller output and plant input and all data to be transmitted needs therefore to be quantized. This introduces a quantization constraint into the problem of minimizing \(V(\bar {u}\,'\!,x_{t})\).
We note that Ψ is fixed and we may at this point either directly quantize \(\bar {u}^{*}\) or instead quantize ξ _{ t } and then apply the mapping Ψ ^{−1} in order to obtain the quantized control vector.^{1} Since Ψ is invertible, and we are transmitting the entire quantized control vector, the resulting coding rate is not affected by this operation [30].
where n _{ t } and ξ _{ t } are mutually independent and ξ _{ t }=Γ x _{ t }. We note that \(\vec {u}_{t}\) is the quantized (and reconstructed) control signal, which has been found by using an ECDQ on ξ _{ t }. Thus, \(\vec {u}_{t}\) is a continuous variable whereas \(\tilde {u}_{t} = \Psi ^{-1}\xi _{t}'\) is the corresponding discrete valued variable, which is entropy coded and thereby converted into a bit-stream (to be transmitted over the network), see Fig. 1. Throughout this work, we will use u _{ t }(i) to refer to the ith element of the vector \(\vec {u}_{t}\).
2.5 MD coding for PPC
We design the MDs by explicitly exploiting the layered construction of the control signals. In particular, we first generate a quantized control vector based on the principles of PPC. This vector contains the current control signal and N−1 future control signals. Then, we construct M descriptions based on this control vector. The descriptions are constructed so that the current control signal and J−1 future control signals can be obtained by combining any subset of \(J\in \{1, \dotsc, M\}\) descriptions. Thus, the more packets that are received at the plant, the more future plant predictions become available. Note that on reception of at least one packet out of the M packets, the current quantized control signal can be completely recovered at the plant input side. When receiving and combining more descriptions, the quality of this control signal is not improved. Instead new control signals become available. With this approach, we thus avoid the issue of having to guarantee stability subject to a probabilistic and time-varying accuracy of the control signals. Instead, we can use ideas from quantized PPC, when assessing the stability. A detailed design of the MDs is provided in Section 4.
3 Theoretical analysis of the PPC-MDC scheme
3.1 Markov jump linear system
where \(\bar {f}_{t}=[f_{t}(1),\dotsc, f_{t}(N)]^{T}\in \mathbb {R}^{N\times 1}\) represents the buffer with the control signals to be applied by the actuator at the plant input side. This buffer holds the present and the N−1 tentative future control values. In particular, f _{ t }(1) is the control value to be applied at current time t, and f _{ t }(i) is to be applied at time t+i−1. In addition, there is also a buffer \(\bar {f}'_{t}\) at the plant side, which holds all received packets that are no older than t−N+1 time instances.
Let \(\Delta _{t}\in \mathbb {R}^{N\times N}\) be an indicator matrix with binary elements {0,1} indicating the complete buffer contents of \(\bar {f}'_{t}\) at time t. In particular, if Δ _{ t } has a “1” at entry (i,j), it shows that at least j packets from time t−i+1 have been received and the buffer therefore contains at least \(u_{t-i+1}(1), u_{t-i+1}(2), \dotsc, u_{t-i+1}(j)\). If, in addition, entry (i,j+1)=1, it further means that the buffer also contains u _{ t−i+1}(j+1). To better illustrate the relationship between Δ _{ t } and the buffers \(\bar {f}'_{t}\) and \(\bar {f}_{t}\) consider the following example.
Example 3.1.
\(\square \)
In the following, we will show that the number of distinct difference indicator matrices is finite for bounded N, and that the sequence of difference indicator matrices {Δ t′} is stationary Markov and ergodic. These properties will be helpful in the subsequent analysis.
Lemma 3.1.
with equality if N=M, i.e., if the number of packets is equal to the horizon length. △
Proof.
See Appendix 1.
Lemma 3.2.
The sequence of difference indicator matrices {Δ t′} is stationary Markov and ergodic. △
Proof.
See Appendix 2.
Example 3.2.
\(\square \)
We are now in a position to introduce the main technical result of this section, which shows that the sequence of augmented state variables {Ξ _{ t }} in (17) and the sequence of difference indicator matrices {Δ t′} in (20) are jointly Markovian and form a Markov jump linear system.
Theorem 3.1.
△
Proof.
See Appendix 3.
3.2 Stability and steady state system analysis
At time step t+1, the switching variable jumps from some particular state, say Δ t′=Δ to some state, say \(\Delta _{t+1}'=\tilde {\Delta }\), where it is possible that \(\Delta =\tilde {\Delta }\). Let the number of distinct states be L, see Lemma 3.1. Thus, without loss of generality, we can enumerate the L (not necessarily distinct) pairs of system matrices that are associated with the L states by \(\{(\mathcal {A}(1), \mathcal {B}(1)), (\mathcal {A}(2), \mathcal {B}(2)), \cdots, (\mathcal {A}(L), \mathcal {B}(L))\}\). We note that even though some of the system matrices might be identical, there is a bijection between the state Δ and the index i of the pair of system matrices. Let p _{ i|j }=Prob(Δ t′=i|Δ t−1′=j), i.e., the transition probability due to jumping from state j to state i, where we note that p _{ i|j } is independent of t due to stationarity of the switching sequence, see Lemma 3.2.
In order to assess the stability of the MJLS in (23) and find its stationary first- and second-order moments, we will first introduce some new notation and then directly invoke Proposition 3.37 in [27], which we for completeness^{3} include as Lemma 3.3 below.
Definition 3.1 (Definitions 3.8 and 3.32 in [27]).
Lemma 3.3 (Proposition 3.37 in [27]).
If \(\sigma _{r}(\mathfrak {A}) < 1\), then the system in (23) is MSS.
Remark 1.
Lemma 3.3 shows that there is an upper limit on the spectral radius of the matrix \(\mathfrak {A}\) given by (34) above which the system cannot be stabilized. This matrix \(\mathfrak {A}\) depends on the packet loss rates via p _{ i|j } and on the delays via the different switching matrices \(\mathcal {A}(i), i=1,\dotsc, L\).
The MJLS in (23) is in general not stationary. However, as can be observed from Definition 3.1, if the system is MSS then asymptotically as \(t\to \infty \), its first- and second-order moments do not depend on t. This observation is formalized in ([27] Theorem 3.33), which we include in part below.
Theorem 3.2 ([27] Theorem 3.33).
If the MJLS in (23) is MSS, then it is also asymptotically wide sense stationary (AWSS) and vice versa.
Lemma 3.4 (Proposition 3.37 in [27]).
△
In our case, we note that γ=0 since the external disturbance w _{ t } and the quantization noise n _{ t } both are zero mean. This implies that q _{ i }=0,∀i, in (43).
3.3 Assessing the coding rate of the quantizer
Recall from Section 2.4 that the quantized control vector \(\tilde {u}_{t}\) is obtained by quantizing ξ _{ t } to get the quantized vector ξ t′ and then using \(\tilde {u}_{t}=\Psi ^{-1}\xi _{t}'\). The following result establishes an upper bound on the bit rate required for transmitting ξ t′.
Theorem 3.3.
where \(\sigma ^{2}_{\bar {\xi }(i)|\bar {\xi }(1),\cdots,\bar {\xi }(i-1)}\) denotes the conditional variance of \(\bar {\xi }(i)\) given \((\bar {\xi }(1),\cdots,\bar {\xi }(i-1))\), and where \(\bar {\xi }\) denotes Gaussian random variables with the same first- and second-order moments as the asymptotically stationary moments of ξ _{ t }.
Proof.
See Appendix 4.
Remark 2.
It is straight-forward to extend Theorem 3.3 to the case of M≤N descriptions by considering M (instead of N) subsets of the vector ξ _{ t }. For example, if N=4 and M=3, one could make the split {ξ t′(1),ξ t′(2),(ξ t′(3),ξ t′(4))}, where upon receiving a single description only ξ t′(1) is recovered, receiving any two descriptions makes it possible to recover ξ t′(1) and ξ t′(2), and receiving all M=3 descriptions, the entire vector \(\xi '_{t}(1),\dotsc, \xi '_{t}(4)\) is recovered. \(\square \)
Remark 3.
This makes the upper bound on the bit rate in (45) computable and thereby relevant from a practical perspective. Indeed, we show in the simulation study in Section 5, that the bound in (45) is very close to (only 1 bit above) the resulting operational bit rate.^{4} \(\square \)
4 Practical design of the PPC-MDC scheme
There are many ways to design MD coding schemes, for example, by use of lattice quantization and index assignment techniques [32, 33], frame expansions followed by quantization [34], oversampling and delta-sigma quantization [35], or layered source coding followed by unequal error protection [36, 37]. In this work, we will be using the latter technique, where the source is decomposed into a number of layers and encoded in such a way that upon reception of say k descriptions, all layers up till the kth layer are revealed [36]. In particular, we rely on a common practical implementation of this strategy, which is based on conventional forward error correction (FEC) codes that are applied on the individual source layers [37]. It will be shown that there exists a natural connection between PPC and MD based on FEC codes, in the sense that a quantized control vector \(\tilde {u}_{t}\) with N−1 future predictions, can be split into M≤N “layers”, where each layer contains at least one control value. Then, based on these M “layers”, we construct M packets \(s_{t}(i), i=1,\dotsc, M\), so that upon reception of any k≤M packets, the control signals \(\tilde {u}_{t}(1), \dotsc, \tilde {u}_{t}(k)\) can be exactly obtained at the decoder. Thus, as more packets are received, more information about future predicted control signals will become available at the plant input side.
4.1 Forward error correction codes
Consider an (n,k)-erasure code, which as input takes k symbols \({y_{t}^{k}}=(y_{t}(1),\dotsc, y_{t}(k))\) and outputs n symbols \(\tilde {y}_{t}^{n}=(\tilde {y}_{t}(1),\dotsc, \tilde {y}_{t}(n))\), where n≥k, and where \(y_{t}, \tilde {y}_{t}\) belong to some (yet to be specified) discrete alphabets. With an (n,k)-erasure code, the original k input symbols can be completely recovered using any subset of at least k output symbols. For example, a (3,2)-erasure code may be constructed by letting \(\tilde {y}_{t}(1) = y_{t}(1), \tilde {y}_{t}(2) = y_{t}(2)\), and \(\tilde {y}_{t}(3) = y_{t}(1) \textrm {XOR} y_{t}(2)\), where the XOR operation is performed on, e.g., the binary expansions of y _{ t }(1) and y _{ t }(2). Thus, using any two \(\tilde {y}_{t}(i),\tilde {y}_{t}(j), i\neq j\) both y _{ t }(1) and y _{ t }(2) may be perfectly recovered. This principle extends to any n>k by using, e.g., erasure codes that are maximum distance separable cf. [38].
4.2 Combining PPC- and FEC-based MDs
To further illustrate the usefulness of the above approach, consider the case where M=5 and where the decoder receives three packets say s _{ t }(2),s _{ t }(3), and s _{ t }(5). Then from say s _{ t }(2), we first recover \(\phi _{t}^{(2)}(1)\), which is in fact identical to \(\tilde {u}_{t}(1)\). Then, from say s _{ t }(2) and s _{ t }(3), we then recover \(\phi _{t}^{(2)}(2)\) and \(\phi _{t}^{(3)}(2)\) from which we can decode \(\tilde {u}_{t}(2)\). Finally, using all three received packets, we recover \(\phi _{t}^{(2)}(3), \phi _{t}^{(3)}(3)\), and \(\phi _{t}^{(5)}(3)\), which can be uniquely decoded to obtain \(\tilde {u}_{t}(3)\).
The foregoing discussion shows that the presence of packet dropouts together with the use of MDs makes the length of the received control packets stochastic and time-varying, while the prediction horizon N is fixed. This aspect makes the analysis of the resultant NCS significantly more involved than that of earlier PPC schemes, as presented in [11]. For example, the number of switching states L, as given by Lemma 3.1, grows exponentially in the horizon length N, whereas in [11] it was enough to consider only two states irrespective of the horizon length.
4.3 Buffering and reconstruction of control signals
Control value \(\hat {u}_{t}\) at time t from available buffer contents
\(\hat {u}_{t}\) | s _{ t }(1) | s _{ t }(2) | s _{ t }(3) | s _{ t−1}(1) | s _{ t−1}(2) | s _{ t−1}(3) | s _{ t−2}(1) | s _{ t−2}(2) | s _{ t−2}(3) |
---|---|---|---|---|---|---|---|---|---|
u _{ t }(1) | 1 | x | x | x | x | x | x | x | x |
u _{ t }(1) | x | 1 | x | x | x | x | x | x | x |
u _{ t }(1) | x | x | 1 | x | x | x | x | x | x |
u _{ t−1}(2) | 0 | 0 | 0 | 1 | 1 | x | x | x | x |
u _{ t−1}(2) | 0 | 0 | 0 | 1 | x | 1 | x | x | x |
u _{ t−1}(2) | 0 | 0 | 0 | x | 1 | 1 | x | x | x |
u _{ t−2}(3) | 0 | 0 | 0 | x | 0 | 0 | 1 | 1 | 1 |
u _{ t−2}(3) | 0 | 0 | 0 | 0 | x | 0 | 1 | 1 | 1 |
u _{ t−2}(3) | 0 | 0 | 0 | 0 | 0 | x | 1 | 1 | 1 |
4.4 Quantization and coding rates
In order to construct the MDs, we need to split the quantized control vector into individual components. It is therefore not possible to directly quantize the vector ξ _{ t } by use of vector quantization as we have done in our previous work on NCS [11], which did not include the use of MDs. Instead, we will in this work use a scalar quantizer separately along each dimension of the vector ξ _{ t }. Of course, a scalar quantizer is not as efficient as a vector quantizer, but the gap from optimality, which is given by \(N/2\log _{2}(\pi e/6)\), is included in the upper bound in (52). Interestingly enough, we can still do vector entropy coding by making use of conditional entropy coding. In particular, we first entropy code the first element of the quantized control vector, i.e., \(\tilde {u}_{t}(1)\). This results in an average discrete entropy of \(H(\tilde {u}_{t}(1)|\zeta _{t})\). Next, we conditional entropy code the second element \(\tilde {u}_{t}(2)\), which results in an average entropy of \(H(\tilde {u}_{t}(2)|\tilde {u}_{t}(1),\zeta _{t})\). This procedure is repeated for the entire vector \(\tilde {u}_{t}\). The FEC code is now applied on outputs of the conditional entropy coders following the approach described in Section 4.2.
As pointed out in Section 2.4, we transmit the elements of \(\tilde {u}_{t}\) and not those of ξ t′. The reason for this is that if we receive ξ t′(1) for the case of N>1, then we are actually not able to reconstruct \(\tilde {u}_{t}(1)\), since \(\tilde {u}_{t} = \Psi ^{-1} \xi _{t}'\). Thus, \(\tilde {u}_{t}(1)\) depends upon the whole vector ξ t′ and not just the first element. Since Ψ ^{−1} is fixed and full rank, it simply maps elements one from discrete set into another discrete set. Thus, the coding rate is not affected by sending \(\tilde {u}_{t}(i)\) instead of ξ t′(i).
Since we have M of these packets, i.e., we have M descriptions, the resulting coding rate is RM.
5 Simulation study
We will now use the analysis and design presented in Sections 3 and 4 in a simulation study in MATLAB.^{6}
5.1 System setup
where the absolute values of the eigenvalues of A are {1.9829,1.2265,1.2265,0.9455,0.9455}. Thus, the system is open-loop unstable. We let the external disturbance \(w_{t}\in \mathbb {R}^{2}\) in (1) be Gaussian distributed with zero mean and covariance matrix Σ _{ w }=I _{2}, where I _{2} denotes the 2×2 identity matrix. The remaining constants in (1) are set to B _{1}=1 _{ z } and B _{2}=[B _{1},B _{1}]. In these simulations, we have used T=4×10^{6} vectors each of dimension z=5 in the sequence \(\{x_{t}\}_{t=0}^{T}\) in (1). x _{0} is initialized to the zero vector.
5.2 Cost function
5.3 Horizon length and number of packets
We consider the cases where N=1,2,3 and compare the proposed scheme that includes multiple descriptions, with the same scheme without multiple descriptions, i.e., that of our earlier work [11]. The two schemes are hereafter referred to as PPC-MDC and PPC, respectively. For the case of PPC-MDC, we let the number of packets M be equal to the horizon length N. For the case of PPC, the entire N-horizon vector is encoded into a single packet. For the case of N=1, the two schemes are identical.
5.4 Network
To simplify the simulations and to be able to compare to existing works on PPC, we will not consider delayed or out-of-order packets. Specifically, if at time t, packet s _{ t−ℓ },ℓ>0 is received, it is discarded. This means that for the case of N=M=3, the number of jump states reduces to L=4 instead of L=196 as given by Lemma 3.1. Note that even though we do not consider late packet arrivals, control signals can still be applied out of order. To see this, assume that M=N=3, and that all three packets {s _{ t }(1),s _{ t }(2),s _{ t }(3)} are received at time t. Then, at time t+1, a single packet is received, say s _{ t+1}(1), and at time t+2 no packets are received. Then, the control signal u _{ t+1}(1) applied at time t+1 is constructed later than the control signal u _{ t }(3) to be applied at time t+2.
We let the packet losses be mutually independent and identically distributed with probability p that a packet is lost (erased). For this case, the state transition probabilities are given by (22).
5.5 Stability
5.6 Quantization
Each scalar control value in the control vector \(\bar {u}_{t}\) is quantized using a uniform scalar quantizer with some step size δ. Specifically, for the case of PPC, we simply keep the step size fixed at δ=10. On the other hand, for the case of PPC-MDC, we need to use a larger step than what is used for PPC, since PPC-MDC introduces redundancy across the M=N descriptions. Thus, to keep the bit rate from growing too much as a function of N, we have experimentally found that δ=25N ^{2} to be a suitable choice, i.e., δ=25,100,225, for N=1,2,3, respectively.
5.7 Bit-rates
To estimate the bit-rate of the quantized control signals, we use (47), which require the computations of discrete conditional entropies. To estimate these conditional entropies, we use a histogram-based entropy estimation on the sequence of discrete (quantized) control signals \(\{\tilde {u}_{t}\}_{t=0}^{T}\). Specifically, we first estimate \(H(\tilde {u}_{t}(1))\) directly from \(\{\tilde {u}_{t}(1)\}_{t=0}^{T}\). Then, we estimate \(H(\tilde {u}_{t}(1), \tilde {u}_{t}(2))\) from \(\{\tilde {u}_{t}(1),\tilde {u}_{t}(2)\}_{t=0}^{T}\) and use that \(H(\tilde {u}_{t}(2)|\tilde {u}_{t}(1)) = H(\tilde {u}_{t}(1), \tilde {u}_{t}(2)) - H(\tilde {u}_{t}(1))\). We obtain \(H(\tilde {u}_{t}(3)|\tilde {u}_{t}(1),\tilde {u}_{t}(2)) = H(\tilde {u}_{t}) - H(\tilde {u}_{t}(1),\tilde {u}_{t}(2))\) in a similar way. Finally, these estimates of the conditional entropies are inserted into (47) in order to approximate the resulting operational bit-rate R. The resulting total discrete entropy R _{ T }=R M obtained by adding the entropies of the M descriptions is shown in Fig. 5 as a function of the packet loss rate and M=N.
It may be noticed in Fig. 5 that the upper bound (45) is approximately 1 bit above the estimate R _{ T } of the operational bit-rates except in the region, where the packet loss rates approach and exceed the critical point, where the system becomes unstable. This excess 1 bit accounts for the theoretical loss of an entropy coder. While we have not applied actual entropy coding, it is well known that the loss of the entropy coder diminishes at moderate to large bit rates.
Note that the multiple descriptions of PPC-MDC have a certain amount of controlled redundancy, and one might therefore expect that the total coding rate for all M=N descriptions would be much greater than what is used for the single description in PPC. However, due to being a closed-loop system, packet losses affect the variance of the input to the quantizer. Consequently, the resulting coding rate for PPC as well as for PPC-MDC also depend upon the packet loss rate.
5.8 Performance
5.9 Complexity
From the analysis of the MJLS in Section 3, it is not easy to assess the computational burden required, when using the proposed system in practice. In this section, we provide a brief overview of the complexity of the encoder and decoder. The encoder includes the controller, quantizer, entropy coder, and channel (FEC) coder. The decoder includes channel decoder, entropy decoder, buffering, and selection of the control values:
Encoder
- 1.
At any given time, say t, the control vector \(\bar {u}_{t}\) is constructed as in (10) and (13), which amount to a few matrix vector multiplications. The matrices in question are \(\Gamma \in \mathbb {R}^{N\times z}\) and \(\Psi \in \mathbb {R}^{N\times N}\), where N is the horizon length and z is the state dimension. For many applications, both the horizon length and the state dimension are moderately small.
- 2.
Each scalar element in either the control vector \(\bar {u}_{t} \in \mathbb {R}^{N}\) or in \(\xi _{t} \in \mathbb {R}^{N}\) is quantized using a scalar quantizer as described in Section 5.6. This amounts to N simple rounding operations, which can be done efficiently in hardware.
- 3.
The quantized elements are entropy encoded either independently, conditionally, or jointly. In either case, it is done in practice by look-up tables and is therefore of low complexity, i.e., \(\mathcal {O}(N)\).
- 4.
The resulting bitstream after entropy coding is converted into M packets by applying M FEC codes, which amounts to matrix-vector multiplications over finite fields [40]. If we use M=N packets, and thereby split the control vector into N “layers”, then the ith layer uses i×N multiplications due to the (N,i) FEC code. Thus, the total number of multiplications is \(N\times (1+2+\cdots + N) = \mathcal {O}(N^{3})\).
Decoder
- 1.
At the decoder at time t, all received packets that are no older than time t−N+1, are stored in a buffer. Moreover, all decoded control values that are no older than time t−N+1 time delays are stored in another buffer. Thus, since there can be M≤N packets in each time slot, the storage complexity is \(\mathcal {O}(MN)\).
- 2.
Decoding of received packets involves decoding the FEC code and decoding the entropy code. Decoding the FEC code can be done by, e.g., Gaussian elimination, which has complexity \(\mathcal {O}(N^{3})\) per layer, and therefore at most \(\mathcal {O}(N^{4})\) for decoding the entire control vector. Decoding of the entropy code is done by a look-up table and has, thus, complexity \(\mathcal {O}(N)\), since the control vector contains N elements.
- 3.
If the decoded control signals are stored in U _{ t } (18), then the selection of the control signal from the buffer can be done as suggested in (19). This includes construction of the vector \(\tilde {\delta }_{i}\) in addition to forming the inner product of \(\tilde {\delta }_{i}\) and the diagonal of U _{ t } indexed by γ _{ i }. The inner product has complexity \(\mathcal {O}(N)\).
6 Conclusions
We have shown how to combine multiple description coding with quantized packetized predictive control, in order to get a high degree of robustness towards packet delays and erasures in network control systems. We focused on a digital network located between the controller and the plant input. In our scheme, when any single packet is received, the most recent control value becomes available at the plant input. Moreover, when any J out of M packets are received, the most recent control value and J−1 future predicted control values become available at the plant input. These future-predicted control values can then be applied at time instances, where no packets are received. The key motivation for this design was twofold. From a practical point of view, it was shown that a significant gain over existing packetized predictive control was possible in the range of large packet loss rates. Moreover, from a theoretical point of view, computable guarantees for stability and upper bounds on the operational bit rate could be established. Indeed, a simulation study revealed that the upper bounds on the bit rate was a good indicator for the operational bit rate of the system in the range of packet loss probabilities that were not too close to the region of system instability.
Future works could include source coding in the feedback channel as well as the forward channel, which is a non-trivial extension. Indeed, the design and analysis of optimal joint controller, encoders, and decoders in both forward and backward channels is an open problem even in the absence of erasures and delays. The main difficulty is that the design of the source coder in the forward channel hinges heavily on the design of the source coder in the backward channel as well as on the controller. Another interesting open research direction is to establish lower bounds on the bit rates, which will then make it possible to assess the optimality of the overall system architecture from an information theoretic point of view.
7 Endnotes
^{1} For the case of quantized MPC with fixed-rate quantization and without dithering, it was shown in [41], that the optimal quantized control vector is given by nearest neighbour quantization of ξ _{ t } in (10).
^{2} It follows that we require the dither sequence to be known both at the encoder and at the decoder.
^{3} We will explicitly make use of (34) – (40) and Lemma 3.3, when assessing the stability of the system in the simulation study in Section 5.
^{4} The excess 1 bit is due to the conservative estimate of the loss of the entropy coder, which is characterized by 1 bit.
^{5} If they are not of equal length, it is always possible to augment one of the sub-bitstreams with a fixed (known) bit pattern to make them of equal length.
^{6} Matlab code to reproduce all results (figures and tables) will be made available online on the authors webpage.
^{7}Of course, information about what time instances the packets were received can be learned from past Δ’s. However, we are not exploiting this knowledge here.
8 Appendix 1: Proof of Lemma 3.1
The case of M<N follows easily from the above analysis. In this case, each row of Δ _{ t } can only take on M+1 distinct patterns, i.e., the zero vector, or a vector containing the number of consecutive ones corresponding to the number of control values that are recovered, when receiving J out of the M packets, where \(J=1,\dotsc,M\). It follows immediately that the number of possible difference indicator matrices is less for M<N compared to M=N.
\(\square \)
9 Appendix 2: Proof of Lemma 3.2
We first prove ergodicity. Clearly, from the all zero difference indicator matrix, it is possible to get to any other difference indicator matrix in a finite number of steps. Moreover, the probability of not receiving any packets in N consecutive time steps is positively bounded away from zero for any finite N. The all zero difference indicator matrix can therefore be reached in a finite number of steps (from any other difference indicator matrix). Thus, it is possible to jump between any two difference indicator matrices in a finite number of steps. We may therefore view the difference indicator matrices as being the different nodes in a fully connected graph. In this graph, any node can be reached at irregular times. Thus, the nodes are recurrent and aperiodic, which implies that they are are ergodic and the sequence {Δ t′} of difference indicator matrices is therefore also ergodic.
Finally, the stationarity of the channel, see Section 2.3, implies that the sequence of difference matrices {Δ t′} is stationary. This proves the lemma. \(\square \)
10 Appendix 3: Proof of Theorem 3.1
In Lemma 3.2, we have established ergodicity and Markov properties of the switching sequence {Δ t′} as is required by Lemma 3.3. We then need to derive the recursive form for the system evolution, which guarantees that the combined system {Ξ _{ t },Δ t′} will be Markovian.
Using that \(\Xi _{t} = \left [\begin {array}{c} \bar {x}_{t} \\ \bar {f}_{t} \end {array}\right ]\) and combining (51) and (50) and using the matrix definitions in (24) – (33) yields (23). This proves the theorem. \(\square \)
11 Appendix 4: Proof of Theorem 3.3
where the last equality follows from the definition of differential entropy of a Gaussian variable [30]. Inserting (58) into (54) and (52) yields (45). This completes the proof. \(\square \)
Declarations
Acknowledgements
This research was partially supported by VILLUM FONDEN Young Investigator Programme, Project No. 10095. The authors would like to thank the reviewers for pointing us to references [23–25] and to [6, 7].
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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