Model Predictive Control of Linear Systems over Networks with State and Input Quantizations

Although there have been a lot of works about the synthesis and analysis of networked control systems (NCSs) with data quantization, most of the results are developed for the case of considering the quantizer only existing in one of the transmission links (either from the sensor to the controller link or from the controller to the actuator link). This paper investigates the synthesis approaches ofmodel predictive control (MPC) for NCS subject to data quantizations in both links. Firstly, a novel model to describe the state and input quantizations of the NCS is addressed by extending the sector bound approach. Further, from the new model, two synthesis approaches of MPC are developed: one parameterizes the infinite horizon control moves into a single state feedback law and the other into a free control move followed by the single state feedback law. Finally, the stability results that explicitly consider the satisfaction of input and state constraints are presented. A numerical example is given to illustrate the effectiveness of the proposed MPC.


Introduction
Model predictive control (MPC) commonly refers to a class of computer control algorithms that use an explicit process model to forecast the future response of a plant [1].In practical applications, MPC can now be found in a wide variety of industrial fields, such as chemical industries, food processing, and automotive applications, since it has advantages including good tracking performance, physical constraints handling, and extension to nonlinear systems.Not only that, in recent decades, it also has a great improvement in theoretical research; various methodologies have been addressed and developed, such as dynamic matrix control (DMC), generalized predictive control (GPC), and robust MPC.From the 1990s, the main stream of the theoretical research about MPC has been turned to the synthesis approach, which means that the MPC controller is synthesized so that the closed-loop system is stable whenever the optimization problem is feasible.A simple procedure of synthesis approach of MPC is to parameterize the infinite horizon control moves into a state feedback control law at each sampling time.In this field, [2] introduces the linear matrix inequality (LMI) technique to solve the optimization problem of minimizing an upper bound on "worst case" value of the infinite horizon objective function and reduces the problem to a convex optimization problem involving a set of LMIs.Reference [3] applies the technique in [2] off-line, and a sequence of control laws corresponding to a sequence of asymptotically stable invariant ellipsoids is constructed.The real-time state feedback gain is chosen from the designed control laws.In this way, the on-line computational burden can be significantly reduced.Another procedure of synthesis approach of MPC is to add free control moves before the linear state feedback law, instead of a single state feedback.In this field, the related works can be found in [4,5].Reference [4] proposes a "quasi-min-max" MPC algorithm for linear parameter varying systems by adding one free control move, and the conservatism and feasibility in [2] are reduced and improved.Reference [5] extends the results in [4] to  > 1 free control moves, which further improves the optimality and feasibility.
Networked control systems (NCSs) are distributed systems in which the sensors, the controllers, and the actuators are spatially distributed and interconnected through communication networks [6][7][8][9][10][11][12]  in NCSs, which is caused by / and / conversions and has an undesirable effect on system performance or even stability.The issue of mitigating the effect of data quantization has been considered by many researchers [13][14][15][16][17][18][19][20].For static and time-invariant quantizer, [13] first points out that the coarsest quantizer should follow a logarithmic law in quadratic stabilization of single-input-single-output (SISO) linear time-invariant (LTI) systems.Reference [14] generalizes the results of [13] to the multi-input-multi-output (MIMO) systems and the output feedback control and converts the quantized quadratic stabilization problem into the robust control problem by using the sector bound method.By adopting a quantization error dependent Lyapunov function, [15] proposes a new approach to the problem of analysis and synthesis for quantized feedback system based on the results in [14], which offers less conservative results.Based on the results in [14], a robust MPC strategy is proposed to deal with the stabilization of NCS with quantized control input in [16] and a simple dynamic scaling method for a logarithmic quantizer based output feedback controller is given in [17].However, all the mentioned papers consider only one quantizer existing in one communication link, either in the sensor to controller (S-C) link or the controller to actuator (C-A) link.The quantized stabilization problem in both links has not received much attention than what it deserves.In the existing results, to the best of our knowledge, there are only a small amount of results [18][19][20] about the quantization problem when both the state and the input are quantized, and all of them do not consider the synthesis approach of MPC.This situation motivates our study presented here.The synthesis approach of MPC is known as a control algorithm that can significantly improve the control performance, especially when system uncertainties exist; see [2,3].Therefore, it is of theoretical and practical significance to extend these approaches to the networked control.However, very few works have focused on the synthesis approaches of MPC for NCS with quantizations.The goal of this paper is to give synthesis approaches of MPC for discrete LTI MIMO system over networks with state and input quantizations.Firstly, by applying the sector bound approach in [14], a novel model which describes the NCS with quantizers in both S-C and C-A links is established.The result converts the quantized stabilization problem into robust control problem.Secondly, based on the developed new model, two synthesis approaches of MPC, without free control move and with one free control move, are derived, respectively, by extending the results in [2,4].Finally, the stability results of the closed-loop system are given by applying the Lyapunov method.
Notation. is the identity matrix with appropriate dimensions.For any vector  and matrix , ‖‖ 2  :=  T .The superscript T denotes the transpose for vectors or matrices.( +  | ) is the value of vector  at a future time  +  predicted at time .

System Description
Consider the quantized feedback control system in Figure 1.The plant to be controlled is modeled by where () ∈ R  and () ∈ R  are the control input and measurable state of the plant, respectively.() ∈ R  and V() ∈ R  are input and output of the controller, respectively.We assume that [() | ()] varies inside a corresponding polytope Ω whose vertices consist of  local systems matrices; that is, with where Co{⋅} denotes the convex hull.The input and state constraints are where The quantized state feedback controller is given by where  ∈ R × is the feedback gain to be designed and (⋅) and (⋅) are two static logarithmic quantizers which are defined as where (⋅) and (⋅) are assumed to be symmetric, that is, In terms of [14], the set of quantization levels for logarithmic quantizers (⋅) and (⋅) is described by where  is  or .The associated quantizer  is defined as follows: with Denote by ♯[] the number of quantization levels in the interval [, 1/].The density of the quantizer  is A small   means coarse quantization and a large   implies dense quantization.In the following, we will call   the quantization density of the quantizer , instead of   , and assume that   is a known number and, hence,   is known and satisfies 0 <   < 1.

Stabilization Using Networked MPC
3.1.Optimization Problem for Networked MPC.As in the nonnetworked MPC (see, e.g., [2,4]), the state feedback V = w will be utilized, where  is to be optimized.We will give two different techniques: case 1 that is without free control move; case 2 that is with one free control move.For simplicity, denote where with Let C = {C ℎ : ℎ ∈ N 2 } be the set of  ×  diagonal matrices whose diagonal elements are either The objective of case 1 is to design () which drives (18) to the equilibrium point  = 0, by minimizing the infinite horizon worst case performance index at each sampling instant: min ) , ( 18) , ( 19) , The objective of case 2 is to design {V(), ()} which drives (24) to the equilibrium point  = 0, by minimizing the infinite horizon worst case performance index at each sampling instant: min ) , ( 19) , (24) . (25) Problems ( 21) and (25) are the "min-max" optimization problem, for which the techniques of LMI can be efficiently utilized.
Proof.Firstly, we consider the recursive feasibility.Assume that the optimization problem ( 38) is feasible at time .The only LMI in this problem which depends on the measured state of the system is (28).Hence, we only need to prove that (28) is feasible for future measured state ( + ),  > 0. From [2], when the constraints (28) and ( 30) are satisfied, it must have At time , the state prediction for  + 1 is At time  + 1, there exist From ( 40) and (41), it is easy to show that ( + 1 |  + 1) is contained in ( + 1 | ).Taking into account (39), we get ( +  |  + 1) T  −1 ( +  |  + 1) < 1, which is equivalent to Thus, the optimization problem is also feasible at time  + 1, and the same results can be achieved at times  + 2,  + 3, . . .by analogy.Hence, the recursive feasibility of optimization problem (38) is guaranteed.Secondly, we prove the asymptotic stability.Assume that the optimization problem (38) is feasible and the solution is depicted as " * ".By applying stability constraints, we have Since (+1 | +1) is contained in (+1 | ), the following holds: By noting it is admissible to choose Solving the optimization problem (38) at time  + 1, it leads to It must have Therefore,  * () decreases monotonously and can be chosen as the Lyapunov function for proving the stability of the closed-loop system.
Mathematical Problems in Engineering 3.3.Networked MPC with One Free Control Move.In Section 3.2, we developed the synthesis approach of networked MPC by parameterizing the infinite horizon control moves into a single state feedback law.This part, in order to reduce the conservatism, provides a synthesis approach of MPC by adding one free control move, which is easier to be feasible and can obtain more optimal control moves.
According to (24), (55) is equivalent to Considering (19) and the convexity of the polytopic description, it is shown that (56) can be guaranteed by Therefore, the "min-max" optimization problem (25) for the proposed quantized networked MPC with one free control move can be approximated by min , 1 ,V(),,,,Γ,  1 +  s.t.
(58) Theorem 3. If the optimization problem (58) is feasible at time , then it is feasible for all times  > , and the receding horizon quantized state feedback control move () = (V(())) asymptotically stabilizes the closed-loop system.
Proof.Assume that the optimization problem (58) is feasible and the optimal solutions are depicted as " * ."At time +1, the feasible solution can be chosen as We can take It is easy to show  ( + 1) ≤  ( + 1) .
At time +1, the optimization is performed again, and it leads to Hence,  * () is decreased monotonously, and we can choose it as the Lyapunov function for proving the stability of the closed-loop system.This completes the proof.

Numerical Example
In this section, the effectiveness of the proposed MPC is illustrated by a numerical example.Consider the following linear parameter varying system:  Let  true = ∑ ∞ =0 (() T () + () 2 ) and denote  ave as the average computation time for each .By solving (38) and (58), the resultant state trajectories, the control input, and the evolution of () (()) are shown in Figures 2, 3, and 4, respectively, in dash lines (solid lines), with  true = 19.005and  ave = 0.2078 s ( true = 18.586 and  ave = 0.2954 s).It is shown that the closed-loop system is stable by applying the two techniques we proposed.Compared with the networked MPC without free control move, by adding one free control move one can achieve faster state responses and more aggressive control moves.Moreover, () and () are monotonic with the evolution of time.

Conclusions
In this paper, the synthesis approaches of MPC have been introduced to deal with the problem of NCS with input and state quantizations.The issue of quantized stabilization of this system is transformed into the robust control problem by applying the sector bound approach.Two synthesis approaches of MPC are formulated and the stability results for the closed-loop system are given by taking the presented  MPC which explicitly consider the input and state constraints.The numerical example illustrates the effectiveness of the proposed techniques.