Robust Tube-Based MPC with Piecewise Affine Control Laws

and Applied Analysis 3 After the basic definitions of polytope have been depicted, the related operations involved in the remaining part of the paper are listed below. Definition 8. TheMinkowski sumof two given setsA andB is defined asA⊕B ≜ {a+ b | a ∈ A, b ∈ B}, and the Minkowski difference is A ⊖ B ≜ {a | a ⊕ B ⊆ A}. Definition 9. The set difference of two setsA andB is defined as A \ B ≜ {a ∈ R n | a ∈ A, a ∉ B} . (8) Definition 10. Given a set A, the affine map of A by the mapping Γ : a 󳨃→ Aa + c, where Γ : Rn → R, to a set B, then B is defined as B ≜ {Aa + c ∈ R m | a ∈ A} , (9) whereA ∈ Rm×n and b ∈ Rm are constant matrix and vector, respectively. Remark 11. Another problem we often faced is to find a polytope A satisfying A ≜ {a ∈ R n | Aa + c ∈ B} , (10) that is, to compute a polytopeA which is a map to B. It is the inverse problem of Definition 10 and can be easily computed by using the affinemap in Definition 10. A function “range” is provided by Matlab toolbox called MPT [31] to compute the B in Definition 10 and a function “domain” for calculating A in (10). Due to the fact that all the variables in this paper are bounded by the polytopes, the state, input, and disturbance constraint sets are assumed to have the following form: X ≜ {x ∈ R n | H x x ≤ K x } , U ≜ {u ∈ R m | H u u ≤ K u } , W ≜ {w ∈ R m | H w w ≤ K w } , (11) where H x ∈ Rhx, K x ∈ Rnhx , H u ∈ Rhu, K u ∈ Rnhu , H w ∈ Rhw, andK w ∈ Rnhw . 3. Multiparametric Quadratic Programming (mp-QP) Themultiparametric programming is the linear or nonlinear programming with parameters in the objective function or/and constraints.The semi-infinite programming andmultilevel programming can be cast as special multiparametric programming. The main advantage of multiparametric programming is that the explicit piecewise affine solutions with respect to the parameters can be calculated. In this section, the general formulation of multiparametric programming is introduced. We refer the readers to [11–19] and the references therein for the basic definitions and results of the multiparametric programming used in this paper. The general formulation of multiparametric programming is as follows: min X f (X, Y) , (12a) s.t. c i (X, Y) ≤ 0, ∀i = 1, . . . , p, (12b) c j (X, Y) = 0, ∀j = 1, . . . , q, (12c) Y ∈ Y ⊆ R ny , (12d) where X ∈ Rnx is the optimization variable and Y ∈ Rny is the parameter variable. Define the Lagrangian as L (X, Y, λ, γ) = f (X, Y) − n


Introduction
The concept of "tube" in model predictive control (MPC) is proposed by [1] for linear discrete-time systems with bounded additive disturbance.The core idea is to design a robust invariant set for the corresponding nominal system, and then the optimization problem with tightening constraints is solved to get the optimal control action [2].Since all the predictions of state and control variables are confined to the tighten constraint set, the real dynamics of controlled systems will never deviate the constraints under the impaction of external disturbances.After the proposition of the algorithm, it has been greatly developed.In [3], the tube-based MPC algorithm is extended to the case of linear sampled-data systems, and in [4] the tracking problem is considered.In [5], the tuning parameter is incorporated into the basic MPC strategy which enables us to move smoothly from the existing controller to a better MPC strategy, and the methodology is applied to the tube-based output-feedback MPC case.A probabilistic tube for linear systems with probabilistic disturbances is designed in [6], which avoids the computation burden in traditional stochastic MPC.A homothetic tube MPC synthesis method is proposed by [7], which utilizes a more general parameterization of the state and control tubes based on homothety and invariance.The tube-based MPC algorithms for nonlinear systems are presented in [8][9][10].In [8], a general nonlinear finite horizon optimization problem with terminal zero constraints is resolved offline once, and the optimal sequence of states and inputs are taken as the reference trajectories, then the on-line optimization problem is designed to tracking these trajectories with constraints satisfaction.In [9], the nonlinear models are locally linearized, and the errors between the linearized models and the true models are confined to lie in the predesigned robust tubes.In [10], a tube-based MPC algorithm for continuous-time nonlinear systems, which satisfies the Lipschitz condition, is proposed.Although the theory of robust tube-based MPC has been stimulated a lot, but far from perfect.
The optimization problems in standard MPC algorithms are in general the linear programming (LP) or linear quadratic programming (QP), and the computation time for LP/QP can not be neglected when the controlled systems have fast dynamics.The explicit MPC takes the LP or QP in standard MPC as multiparametric linear programming (mp-LP) or multiparametric quadratic programming (mp-QP) and solves these optimization problems absolutely offline.
The basic results and development on multiparametric programming can be found in [11][12][13][14] for mp-LP and [15][16][17][18][19] for mp-QP.In this paper, we mainly consider the mp-QP, which is the most general form of optimization we faced in the regular MPC.In [20], the theoretical perspectives of multiparametric programming and explicit MPC are introduced.In [21], the authors present a method to compute the explicit state-feedback control laws for both the MPC algorithm and the constrained linear quadratic regulation problem with guaranteed feasibility, stability, and optimality, in which the explicit feedback control laws are piecewise affine and continuous.In [22], the approximated explicit control laws for MPC are obtained, which utilizes the inner/outer polytopic approximation technique and the implicit doubledescription algorithm.In [23], the systematical procedures for the analytical expression of explicit control laws of linear MPC via piecewise affine function are given, which saves the online computation time and memory requirements.There are many works which have been published on finding ways to resolve the mp-QP.In [17], an efficient mp-QP solver which avoids unnecessary partitioning of the parameter space by directly exploring the neighborhood of initial partition is presented.In [18], an mp-QP solver with a new partitioning method of the parametric (state vector) space is proposed, which avoids the unnecessary partitioning and improves the efficiency.In [19], an algorithm is proposed to revise the existing algorithms in order to make them satisfy the facet-to-facet property in general and guarantee that the entire parameter space is explored.Because the partitions of feasible sets are increasing exponentially with respect to the prediction horizon, the complexity reduction methods are presented by [24][25][26] to remove the unnecessary partitions.The only online calculation for mp-QP in MPC is the point location problem (to confirm which region contains the current parameter).In [27][28][29], different efficient methodologies have been proposed to resolve this problem.
This paper considers the constrained discrete-time linear systems with additive disturbances.The disturbances are assumed to be confined in a polytope.First, the standard tube-based MPC algorithm, which solves the quadratic programming (QP) online and takes initial state in the optimization as an optimized variable, is designed.Then, by transforming the optimization problem in standard tubebased MPC into the mp-QP form, the optimization can be solved through mp-QP solvers, which separate the parameter space into finite partitions and get a piecewise affine linear optimal solution for each partition.In this paper, the standard tube-based MPC algorithm is solved absolutely offline, and the only online calculations are to confirm which partition the current state lies in.
The remainder of this paper is structured as follows.In Section 2, the basic formulation of the controlled systems and basic definitions on polytope and polyhedron are introduced.The multiparametric programming is described in Section 3. In Section 4, the tube-based MPC algorithm with piecewise affine state feedback control laws is presented.In Section 5, the simulation results are provided to show the effectiveness of the proposed algorithm.

Problem Statement and Preliminaries
Consider the following discrete-time linear systems with additive disturbances: where () ∈ R  , () ∈ R  , and () ∈ R  are the state, input, and disturbance vector at sampling time , respectively.
The system states and inputs are constrained by and the disturbance () is bounded by The corresponding nominal model of ( 1) is given as The system (1) is supposed to satisfy the following assumptions throughout the paper.
Assumption 3. X ⊆ R  , U ⊆ R  , and W ⊆ R  are compact and convex and contain the origin as interior point, respectively.
Since all the involved constraint sets in this note are confined as the convex and bounded region, that is, the polytope, the basic definitions related to polyhedron and polytope are shown to make the paper more complete and concise.For more knowledge about the polyhedron and polytope, readers can refer to [30][31][32].Definition 4. A polyhedron R ⊆ R  is a convex set which originates from the intersection of a finite number of halfspaces where  ∈ R × and  ∈ R  and  is a finite integer.
Definition 5. A polytope P ⊆ R  is the bounded polyhedron Definition 6.The H-representation of a polytope P, as in (6), is to depict the polytope P as an intersection region of a finite number of half-spaces.The other way of statement is that the H-representation of a polytope is the region described by a finite number of linear inequalities  ≤ ; that is, the dimensions of  and  are not infinite.
Definition 7. The V-representation of a polytope P is to describe the polytope as a convex hull of its vertices: where  1 ,  2 , . . .,   are the vertices and  is the total number of vertices.
After the basic definitions of polytope have been depicted, the related operations involved in the remaining part of the paper are listed below.
Definition 10.Given a set A, the affine map of A by the mapping Γ :   →  + , where Γ : R  → R  , to a set B, then B is defined as where  ∈ R × and  ∈ R  are constant matrix and vector, respectively.
Remark 11.Another problem we often faced is to find a polytope A satisfying that is, to compute a polytope A which is a map to B. It is the inverse problem of Definition 10 and can be easily computed by using the affine map in Definition 10.A function "range" is provided by Matlab toolbox called MPT [31] to compute the B in Definition 10 and a function "domain" for calculating A in (10).
Due to the fact that all the variables in this paper are bounded by the polytopes, the state, input, and disturbance constraint sets are assumed to have the following form: where

Multiparametric Quadratic Programming (mp-QP)
The multiparametric programming is the linear or nonlinear programming with parameters in the objective function or/and constraints.The semi-infinite programming and multilevel programming can be cast as special multiparametric programming.The main advantage of multiparametric programming is that the explicit piecewise affine solutions with respect to the parameters can be calculated.In this section, the general formulation of multiparametric programming is introduced.We refer the readers to [11][12][13][14][15][16][17][18][19] and the references therein for the basic definitions and results of the multiparametric programming used in this paper.
(, ) = 0, ∀ = 1, . . ., , where  ∈ R   is the optimization variable and  ∈ R   is the parameter variable.Define the Lagrangian as where  ∈ R  and  ∈ R  are the Lagrange multipliers.
Although the solution satisfying the first-order KKT condition may not be the optimal one, for the case in the following theorem, the solution is optimal.
Proof.The proof can be found in [33].
Most of the optimization problems involved in linear control theory can be transformed into the mp-LP or mp-QP.The mp-LP and mp-QP correspond to the case of linear systems with linear performance cost functions and linear systems with quadratic performance cost functions, respectively.In general MPC algorithms, the mp-QP is the most common optimization problem we utilized.
In this paper, we consider the following form of mp-QP problem [17,18,21]: where Then, from (14a), (14b), (14c), (14d), (14e), (14f), and (14g) the first-order KKT conditions of (16a) and (16b) are where , and  ∈ R  .Define  ≜ {1, 2, . . ., }.By analyzing the KKT conditions (17a), (17b), (17c), and (17d), the explicit solution at a given  is given by [18] where 18) is the function of , define Y 0 as the region of  in which the equation of  in (18) remains optimal.The region Y 0 is a polytope and its H-representation is as follows [18]: where To solve the mp-QP problem (16a) and (16b), the feasible region Y  of  needs to be confirmed firstly.Assume that the region Y  ≜ {   ≤   } and is the largest ball contained in Y ; then it can be calculated by solving the following LP problem [21] In order to get the explicit piecewise affine solutions on the whole feasible set Y  , the feasible set needs to be partitioned into a finite number of polytopes.Suppose Y  is partitioned into  partitions; then the partitions P  ,  ∈ {1, 2, . . ., }, have to satisfy , where D  denotes the border of partition P  .Remark 14.The partitions P  ,  ∈ {1, 2, . . ., }, are determined by finding the region Y 0 defined in (19) on the whole Y  .There now exist many methods to determine the partitions P  , such as the method by exploiting the facet-to-facet property [19,21] and adding/withdrawing constraints from active set [18].Theorem 15.Consider the mp-QP of (16a) and (16b) and let Y be a polytope.Then the feasible set Y  ⊆ Y is convex, the optimizer  is continuous and piecewise affine in each partition P  , and the value function   () is continuous, convex, and piecewise quadratic.
Proof.The proof is in [21].
Remark 16.The convexity and continuous properties in Theorem 15 are critical for the optimality of mp-QP.From Theorem 25, since all the involved sets are convex, the inequality constraints in (16b) and the objective function are convex functions in each partition P  ; then the optimizer  is the optimal solution of (16a) and (16b) in the whole region of each partition P  .

Tube-Based MPC with Piecewise Affine Control Laws
In this section, a robust tube-based MPC algorithm with piecewise affine solutions is proposed.For the basic definitions and researching development on the general tube-based MPC algorithm, readers can refer to [1,2].For clarity, the linear nominal model of ( 1) is rewritten as Since the system (22) satisfies Assumption 1, then there exists a feedback gain matrix , such that closed-loop system ( + 1) =   (), where   =  + , is stable.
In the sequel, the basic definition of robust invariant set is introduced by referring to [34].Definition 17.The set Z is said to be a robust invariant set for uncertain linear system ( + 1) =   () + (), if for any (0) ∈ Z and any sequence of disturbances () ∈ W, the solution of ( + 1) =   () + () satisfies () ∈ Z for all  ≥ 0.
Remark 18.In [2], a concise description of the condition is presented as   Z ⊕ W ⊆ Z, where ⊕ denotes the Minkowski sum defined in Definition 8.By assuming that the set W is a polytope, that is, . .,  ℎ }, the robust invariant set is chosen as the minimal robust positive invariant (mRPI) set  ∞ for uncertain linear system ( + 1) =   () + () by the outer -approximation approaches in [35].The main procedures are listed as follows.
Algorithm 19 (the procedures for finding Z).
Step 2. Compute   () by the following equation: where ℎ W () ≜ sup ∈W    and   is the standard basis vector in R  .Set  =   ().
Step 5. Compute F  as the Minkowski sum Proof.The proof is in [1].
For the purpose of proving the stability of tube-based MPC, the general robust MPC algorithm with tighten constraints is described as follows: with the objective function where ( +  | ) is the prediction of  at the future time  + , predicted at time , and ( | ) equals the current state ().The corresponding weighting matrices , , and  in ( 27) are chosen to be positive definite.The tighten constraint set X, U is defined as X ≜ X ⊖ Z, U ≜ U ⊖ Z.The terminal constraint set X  satisfies X  ⊆ X ⊖ Z and contains the equilibrium point in its interior.The optimization (26a), (26b), (26c), (26d), and (26e) can easily be transformed into the QP problem, the solvers for which are fast and efficient in most cases.
Let the terminal cost function   () = ()  () be a Lyapunov function; then the terminal constraints are chosen to satisfy the following stability conditions [2,36]: , ∀() ∈ X  .The principle for choosing X  is to satisfy condition C.1.Then, the terminal region X  can be chosen as the maximum output admissible set  ∞ , as in [37].Consider the linear autonomous system ( + 1) =   () with the output constraint (state constraint) The main procedures to determine X  are listed as follows [37,38].
Algorithm 21 (the procedures for finding X  ).
The -step feasible set X 1  for (26a), (26b), (26c), (26d), and (26e) is defined as Remark 22.The -step feasible set for optimization (26a), (26b), (26c), (26d), and (26e), which is also called the step controllable set, is the set of initial states which can be stabilized to the target set in  steps.The procedures for computing the -step feasible set can be found in [32].
The robust tube-based MPC algorithm, proposed by [1,2], is to solve the following optimization: where the parameters , , , X, U, and X  and the objective function ((), (⋅)) are the same as in optimization (26a), (26b), (26c), (26d), and (26e).In this optimization, the current state () does not equal ( | ), which is the prediction of current state () at time .Z is the robust invariant set solved by Algorithm 19.
In the sequel, we define In spite of the fact that the QP problem can be solved efficiently by the existing solvers, for systems with fast dynamics, the computational burden is still too huge to control these systems well.Even worse, the long computation time may make the controlled systems unstable.A viable way to tackle this problem is to use the offline approach, which solves the related optimization totally offline.The explicit regulator together with multiparametric programming is an alternative way to give a control strategy with less computation time.
The explicit piecewise affine linear control law for robust tube-based MPC at the initial point () ∈ P  is defined as Remark 27.Comparing (44) with (34), the control law (44) is explicit piecewise affine linear with respect to the initial state () and can be computed totally offline.The piecewise affine linear control laws are prestored in the memory of computer, and when these control laws are needed to be applied online, the only measure to take is finding the partition P  which contains () by searching in a lookup table.So the computation time has been extremely decreased, and the algorithm is fit for controlling those systems with fast dynamics.
Since the explicit piecewise affine state feedback control laws are stored in a lookup table, certain searching methods are needed to confirm which partition contains the current state ().The related procedures are referred to as the wellknown "point location" problem.There already exist many viable and efficient methods to figure out these point location problems, such as the binary search tree utilized in [27], the subdivision walking method in [28], and the hash tables in [29].
(i) The feasible set X 3  ⊆ X is convex, the optimizer Ȗ is continuous and piecewise affine in each partition P  , and the value function  3  (()) is continuous, convex, and piecewise quadratic.
(ii) The piecewise affine linear control law  *  (()) in (44) for tube-based MPC algorithm is continuous and piecewise affine in each partition P  .

Illustrative Example
In order to verify the efficiency of the control strategy, the proposed algorithm is applied to a disturbed linear system, which is borrowed from [2].For simplicity and comparison, the related parameters are mostly the same as [2].The system model is described as follows: where The bounds on the state, control, and disturbance are The corresponding nominal model is Choose For the nominal model (49), the optimal state-feedback gain  is calculated by solving the discrete-time LQR problem.By applying the Matlab function "dlqr, "  is calculated as and the corresponding positive symmetric matrix  is  = [ 2.0066 0.5099 0.5099 1.2682 ] .
The robust invariant set Z for (1) under () = () can be computed from Algorithm 19.Z is a polytope and its Hrepresentation is where Similarly, the terminal region X  can be calculated from Algorithm 21.The H-representation of X  is where Choose  = 9 and initial state (0) = (−5, −2).By utilizing the algorithm proposed in this paper, the feasible sets X 1 9 , X 2 9 , and X 3 9 are plotted in Figure 1, the partitions of X 3 9 are shown in Figure 2, and the phase trajectory with embedded robust invariant sets Z from the starting point (0) = (−5, −2) is plotted in Figure 3.In order to test whether the constraints on state and input are violated or not, the Monte Carlo method is utilized and the results are shown in Figures 4 and 5. Also to verify the feasibility in the whole feasible set X 3  9 , different representative initial points are simulated and the corresponding phase trajectories are plotted in Figure 6.Moreover, for comparison, the partitions of the feasible sets X 3  ,  = 1, 3, 5, 7, are plotted in Figures x * (0|0)     7, 8, 9, and 10.The total number of partitions of X 3  ,  = 1, 2, . . ., 9, is shown in Table 1.And the comparison of online computation time between the method in [2] and the proposed method in this paper for each control step is given in Table 2.For simplicity, the representative partitions of X 3 1 and corresponding piecewise affine control laws are listed in Table 3.
Figure 1 shows that X 1  ⊂ X 2  = X 3  .Figure 2 combined with Figures 7-10, or directly Table 1, shows that the number of partitions P  is exponentially increased with respect to the prediction horizon .It can also be seen from Figure 4 that the system (46), under the piecewise affine control laws (44), is asymptotic stable to region 0 ⊕ Z, and all the hard constraints on state and input are not violated at all. Figure 7 illustrates that the state in the obtained feasible set X 3  9 is feasible.The total simulations are achieved using Matlab 7.6a  and the MPT [31] on our laptop with a 2.67 GHz Intel Core i5 processor and 4 GB RAM.

Conclusion
In this paper, a robust tube-based MPC algorithm with piecewise affine control laws is proposed to control the disturbed linear systems.The main advantage of the proposed  algorithm is that the explicit control laws, which are piecewise affine to the partition of feasible set, are obtained totally offline by transforming the optimization of the general tubebased MPC into the mp-QP.Since the involved functions are convex and all the sets are polytopes, the piecewise affine control laws are optimal.By applying these control laws to the controlled system, the stability and robustness properties still hold.The simulation results illustrate the effectiveness.

Definition 8 .Definition 9 .
The Minkowski sum of two given sets A and B is defined as A ⊕ B ≜ { +  |  ∈ A,  ∈ B}, and the Minkowski difference is A ⊖ B ≜ { |  ⊕ B ⊆ A}.The set difference of two sets A and B is defined as

Table 1 :
Number of partitions.

Table 2 :
Mean online computation time for each control step.

Table 3 :
Representative partitions of