Sparse Packetized Predictive Control for Networked Control over Erasure Channels

I Introduction

In networked control systems (NCSs) communication between controller(s) and plant(s) is made through unreliable and rate-limited communication links such as wireless networks and the Internet; see e.g., . Many interesting challenges arise and successful NCS design methods need to consider both control and communication aspects. In particular, so-called packetized predictive control (PPC) has been shown to have favorable stability and performance properties, especially in the presence of packet-dropouts . In PPC, the controller output is obtained through minimizing a finite-horizon cost function on-line and in a receding horizon manner. Each control packet contains a sequence of tentative plant inputs for a finite horizon of future time instants and is transmitted through a communication channel. Packets which are successfully received at the plant actuator side, are stored in a buffer to be used whenever later packets are dropped. When there are no packet-dropouts, PPC reduces to model predictive control. For PPC to give desirable closed-loop properties, the more unreliable the network is, the larger the horizon length (and thus the number of tentative plant input values contained in each packet) needs to be chosen. Clearly, in principle, this would require increasing the network bandwidth (i.e., its bit-rate), unless the transmitted signals are suitably encoded. It is well-known that there exists a minimum bit-rate for achieving stability of a networked feedback control system . The optimal quantizer for the minimum bit-rate is a dynamic vector quantizer, and is, thus, hard to use in many applications. As an alternative, memoryless scalar quantizers, will often be preferable. In this case, sparse representations can be used to reduce the data size of transmitted vectors in PPC. Sparse representations aim at designing sparse vectors, which have few non-zero coefficients, along with optimizing some performance indices. Since sparse vectors contain many zero-valued elements, they can be easily compressed by only encoding a few nonzero coefficients and their locations with a memoryless scalar quantizer. Well-known examples of this kind of encoding are JPEG in image processing and algebraic CELP in speech coding [8, Section 17.11.1]. Over the past few years, a number of studies have been published which deal with sparsity for control, including topics such as trajectory generation , state observation , optimal control , and also sampled-data control .

The purpose of the present work is to introduce sparsity-promoting optimizations for networked control with dropouts. We will show that sparsity-promoting cost functions can be used in PPC to achieve good control performance (as measured by a weighted quadratic norm of the system state), whilst transmitting sequences with only few non-zero elements. By studying the sequence of optimal cost functions at the instances of successful reception, we derive sufficient conditions for (practical) closed-loop stability in the presence of bounded packet-dropouts.

The remainder of this note is organized as follows: Section II revises basic elements of packetized predictive control. In Section III, we show the motivation of sparsity-promoting optimization for PPC, and formulate the design of the sparse control packets. In Section IV, we study stability of the resultant networked control system. A numerical example is included in Section V. Section VI draws conclusions.

II Packetized Predictive Networked Control

Let us consider an unconstrained discrete-time linear time-invariant plant model with a scalar input:

We are interested in an NCS architecture, where the controller communicates with the plant actuator through an erasure channel, as depicted in Fig. 1.

It is worth noting that (1) does not include disturbances. Hence, as an alternative to PPC, one could simply transmit the system state to the actuator and, upon successful reception, the actuator could calculate and implement a semi-infinite plant input sequence. In the present work, we focus on situations where the actuator does not have sufficient computational capabilities precluding such an open-loop control scheme. In contrast, the sparse PPC formulations proposed in the present work provide feedback at all instances where no dropouts occur. Our recent results concerning related schemes, see , suggest that, in the presence of disturbances, PPC will exhibit favorable robustness properties.

III Design of Sparse Control Packets

In the present section we present two methods for the design of sparse PPC. The purpose is to obtain many zero elements $u_{i}({\boldsymbol{x}}(k))$ in the control packet ${\boldsymbol{u}}({\boldsymbol{x}}(k))$, cf., . The control packet ${\boldsymbol{u}}({\boldsymbol{x}}(k))$ is designed at each time $k$ via a standard model predictive control formulation:

Here, $F({\boldsymbol{x}})=0$, whereas the stage cost $L$ is given by $L({\boldsymbol{x}},u)=1$ if $u\neq 0$ and $L({\boldsymbol{x}},u)=0$ if $u=0$. The constraint set ${\mathcal{U}}({\boldsymbol{x}})$, used in (2) is taken as

where $S$ is a positive integer less than $N$. The optimization above may be effectively solved via the CoSaMP algorithm described in . Since the bound $S$ of $\|{\boldsymbol{u}}\|_{0}$ is specified a priori, one can adopt the interleaved single pulse permutation (ISPP) design [8, Section 17.11.1] for effectively encoding the support data of ${\boldsymbol{u}}$. A disadvantage of this approach is the difficulty in estimating a bound $S$ that guarantees stability of the feedback loop. In contrast, in the following section we will show how design parameters in (2) can be chosen to ensure closed loop stability in the presence of bounded dropouts.

IV Stability Analysis of Sparse PPC Loops

The number of consecutive packet-dropouts is uniformly bounded by $N-1$. $\square$

In view of the above, the horizon length $N$ in (2) allows one to trade computational complexity of the on-line optimization for robustness with respect to dropouts. Thus, the less reliable the network is, the larger $N$ should be chosen.

It follows that if ${\boldsymbol{x}}(k)\in\Omega$ and there are no dropouts at time $k$, then the control will be $u(k)=0$. That is, the control system (1) behaves as an open-loop system in the set $\Omega$. Hence, asymptotic stability will in general not be achieved, if $A$ has eigenvalues outside the unit circle. This fundamental property is linked to sparsity of the control vector.

By the fact mentioned above, we will next turn our attention to practical stability (i.e., stability of a set) of the associated networked control system. For that purpose, we will analyze the value function

where $J({\boldsymbol{x}},{\boldsymbol{u}})$ is as in (4). First, we find bounds of $V({\boldsymbol{x}})$.

where $\phi(t)\triangleq a_{1}t+(a_{2}+\lambda_{\max}(Q))t^{2}$, $a_{1}\triangleq\mu\sqrt{n}~\sigma_{\max}\left(G^{\dagger}H\right)$, $a_{2}\triangleq\lambda_{\max}(W^{\star})$, and the matrices $G^{\dagger}$ and $W^{\star}$ are given by

Applying ${\boldsymbol{u}}^{\star}({\boldsymbol{x}})=G^{\dagger}H{\boldsymbol{x}}$ to the cost in (7) gives

the upper bound for $V({\boldsymbol{x}})$ given in Lemma 5 will be tight, if $\mu$ is small. $\square$

Having established the above preliminary results, we introduce the $i$-th iterated mapping ${\boldsymbol{f}}^{i}$ with the optimal vector ${\boldsymbol{u}}({\boldsymbol{x}})=[u_{0}({\boldsymbol{x}}),\dots,u_{N-1}({\boldsymbol{x}})]^{\top}$ defined in (4) through the recursion

This mapping describes the plant state evolution during periods of consecutive packet-dropouts. Note that, since the input ${\boldsymbol{u}}({\boldsymbol{x}})$ is not a linear function of ${\boldsymbol{x}}$ (see Proposition 4), the function ${\boldsymbol{f}}^{i}({\boldsymbol{x}})$ is nonlinear. The following bound plays a crucial role to establish deterministic stability guarantees:

Assume that $P>0$ satisfies the following Riccati equation

Fix $i\in\{1,\ldots,N-1\}$ and consider the sequence

The above result can be used to derive the following contraction property of the optimal costs during periods of successive packet-dropouts:

In this proof, we borrow a technique used in the proof of [37, Theorem 4.2.5]. By Lemma 5, for ${\boldsymbol{x}}\neq{\boldsymbol{0}}$ we have 0 ¡ V(x) ≤a_1 ∥x∥_2 + (a_2 + λ_max(Q)) ∥x∥_2^2. Now suppose that $0<\|{\boldsymbol{x}}\|_{2}\leq 1$. Then $\|{\boldsymbol{x}}\|_{2}^{2}\leq\|{\boldsymbol{x}}\|_{2}$ and hence $V({\boldsymbol{x}})\leq(a_{1}+a_{2}+\lambda_{\max}(Q))\|{\boldsymbol{x}}\|_{2}$. From Lemma 7, it follows that

Since $0<\lambda_{\min}(Q)\leq\lambda_{\max}(Q)$, $a_{1}>0$, and $a_{2}>0$, it follows that $\rho\in(0,1)$.

Next, consider the case where $\|{\boldsymbol{x}}\|_{2}>1$ so that $\|{\boldsymbol{x}}\|_{2}<\|{\boldsymbol{x}}\|_{2}^{2}$ and $V({\boldsymbol{x}})<(a_{1}+a_{2}+\lambda_{\max}(Q))\|{\boldsymbol{x}}\|_{2}^{2}$. This and Lemma 7 give

If ${\boldsymbol{x}}={\boldsymbol{0}}$, then the above inequality also holds since $V({\boldsymbol{0}})={\boldsymbol{0}}$. ∎

Denote the time instants where there are no packet-dropouts, i.e., where $d(k)=0$, as

whereas the number of consecutive packet-dropouts is denoted via:

for $k\in\{k_{i}+1,k_{i}+2,\ldots,k_{i}+m_{i}\}$, and also for $k_{i+1}=k_{i}+m_{i}+1$, we have

Now by induction from (17), it is easy to see that from Lemma 5,

for $k\in\{k_{i}+1,k_{i}+2,\ldots,k_{i+1}-1\}$, and this inequality also holds for $k=k_{i+1}$. Finally, by using the lower bound of $V({\boldsymbol{x}})$ provided in Lemma 5, we have

where $R$ is defined in (13) and we used the inequality $\sqrt{a+b}\leq\sqrt{a}+\sqrt{b}$, for all $a,b\geq 0$. The above inequality leads to (13). ∎

Theorem 9 establishes practical stability of the networked control system. It shows that, provided the conditions are met, the plant state will be ultimately bounded in a ball of radius $R$. It is worth noting that, as in other stability results which use Lyapunov techniques, this bound will, in general, not be tight.

Based on this lemma, we hereafter assume that

The feasible solutions for (6) can be characterized as follows:

The fact $G{\boldsymbol{\varepsilon}}({\boldsymbol{x}})\perp(G{\boldsymbol{u}}^{\star}({\boldsymbol{x}})-H{\boldsymbol{x}})$ gives the result. ∎

The error term ${\boldsymbol{\varepsilon}}({\boldsymbol{x}})$ in (19) may be interpreted as a “penalty charge” for sparsifying the vector (control packet) ${\boldsymbol{u}}$, since the term $\|G{\boldsymbol{u}}({\boldsymbol{x}})-H{\boldsymbol{x}}\|_{2}^{2}$ with the sparse control ${\boldsymbol{u}}({\boldsymbol{x}})$ will be larger than with the least squares one, $\|G{\boldsymbol{u}}^{\star}({\boldsymbol{x}})-H{\boldsymbol{x}}\|_{2}^{2}$. $\square$

Now take $Q>0$ arbitrarily and let $P>0$ be the solution to the Riccati equation (9) with $r=0$. Then, from Lemma 11 and well-known results in dynamic programming [38, Chapter 3], all feasible control vectors ${\boldsymbol{u}}({\boldsymbol{x}})\in{\mathcal{U}}({\boldsymbol{x}})$ can be written as

and $\varepsilon_{i}({\boldsymbol{x}})$ is the $(i+1)$-th element of ${\boldsymbol{\varepsilon}}({\boldsymbol{x}})$ satisfying the inequality in (19). The associated open-loop states are

By using the definition (20) of the matrix $K$, we have

Suppose $Q>0$ is chosen arbitrarily, $P>0$ is the solution of the Riccati equation (9) with $r=0$, and $W>0$ is such that $W>W^{\star}$. Let ${\mathcal{E}}=W-W^{\star}$. Then there exist constants $\rho\in[0,1)$ and $c>0$ such that

Substitution of the state ${\boldsymbol{x}}_{i+1}$ given in (21) into $V_{P}({\boldsymbol{x}})$ yields that V_P(x_i+1) = V_P(x_i) -∥x_i∥_Q^2 + B^⊤PB—w_i(x)—^2. By the definition of $w_{i}({\boldsymbol{x}})$ in (22), we have

where the last inequality is due to Lemma 11, and

Since $P\geq Q>0$, we have $\rho\in[0,1)$. By mathematical induction, we finally obtain

where $\quad c\triangleq(1-\rho)^{-1}(1-\rho^{N})c_{1}$. ∎

Suppose that the matrices $P$, $Q$, and $W$ are chosen by the following procedure:

Solve the Riccati equation (9) with $r=0$ to obtain $P>0$.

Compute $\rho\in[0,1)$ and $c>0$ via (24), (25), and (26).

Choose ${\mathcal{E}}$ such that $0<{\mathcal{E}}<(1-\rho)P/c$.

Compute $W^{\star}=P-Q$ and set $W=W^{\star}+{\mathcal{E}}$.

for $k=k_{i},k_{i}+1,\ldots,k_{i}+m_{i}$. Also, for $k_{i+1}=k_{i}+m_{i}+1$, the next instant when the control packet is successfully transmitted, we have V_P(x(k_i+1)) ¡ V_P(x(k_i+1-1)) ¡ V_P(x(k_i)). It follows that at the time instants $k_{0},k_{1},\ldots$ (no-dropout instants), $0\leq V_{P}({\boldsymbol{x}}(k_{i}))$ strictly decreases, and hence ${\boldsymbol{x}}(k_{i})\rightarrow{\boldsymbol{0}}$ as $i\rightarrow\infty$. Then, by (27), for $k=k_{i},k_{i}+1,\ldots,k_{i}+m_{i}$ (consecutive dropout instants), $V_{P}({\boldsymbol{x}}(k))$ is bounded by $V_{P}({\boldsymbol{x}}(k_{i}))$. Since the latter converges to zero, we conclude that ${\boldsymbol{x}}(k)\rightarrow{\boldsymbol{0}}$ as $k\rightarrow\infty$. ∎

In summary, the networked control system affected by bounded packet-dropouts is asymptotically stable with the sparse control packets obtained by the optimization (6) if $P$, $Q$, and $W$ are computed as per Theorem 14.

V Simulation Study

To assess the effectiveness of the proposed sparse control methods, we consider a plant model of the form (1) with The elements of these matrices are generated by random sampling from the normal distribution with mean 0 and variance 1. Note that the matrix $A$ has 2 unstable eigenvalues ($1.5259$ and $-1.1441$) and 2 stable eigenvalues ($0.4198$ and $-0.7724$).

that minimizes $\|G{\boldsymbol{u}}-H{\boldsymbol{x}}\|_{2}^{2}+r\|{\boldsymbol{u}}\|_{2}^{2}$, and the ideal least squares solution, namely, ${\boldsymbol{u}}^{\star}({\boldsymbol{x}})=G^{\dagger}H{\boldsymbol{x}}.$

VI Conclusions

Future work may include obtaining analytical bounds on the sparsity of solutions. It is also of interest to apply the proposed control methods to constrained nonlinear plant models with disturbances, and to channels with bit-rate limitations and unbounded packet-dropouts. We foresee that this will require extending results in and also the development of fast algorithms to solve the associated optimization problems.

Acknowledgments

The authors wish to thank the Associate Editor and the anonymous reviewers for valuable comments which have helped to improve the quality of this note.

References