P REDICTIVE C ONTROL A DVANCESON D ISTRIBUTED M ODEL

(1)

UNIVERSITÁ DIPISA

DOTTORATO DI RICERCA ININGEGNERIA DELL’INFORMAZIONE

A

DVANCES ON

D

ISTRIBUTED

M

ODEL

P

REDICTIVE

C

ONTROL

DOCTORALTHESIS

Author

Matteo Razzanelli

Tutors

Prof. Mario Innocenti Prof. Lorenzo Pollini Prof. Gabriele Pannocchia

Reviewers

Prof. Giancarlo Ferrari Trecate Prof. Fabrizio Giulietti

The Coordinator of the PhD Program

Prof. Marco Luise

Pisa, May 2019 Cycle XXXI

(2)

(3)

(4)

(5)

"Science is but a perversion of itself unless it has as its ultimate goal the betterment of humanity." Nikola Tesla

"Beh, ecco, se mi posso permettere, spesso il termine utopia è la maniera più comoda per liquidare quello che non si ha voglia, capacità o coraggio di fare. Un sogno sembra un sogno fino a quando non si comincia a lavorarci. E allora può diventare qualcosa di infinitamente più grande" Adriano Olivetti

(6)

(7)

SUMMARY

M

ODEL Predictive Control (MPC) is a class of advanced control

techniques, widely used especially in the process industries, and it has its fundamentals in optimal control. Further, several class of predictive controllers were developed in the last two decades. Neverthe-less, conventional feedback controllers (e.g., PID) are, so far, the de-facto standard for most industrial applications. This is due to the fact that no system model is necessary for these approaches. On the other hand, in case of large-scale applications, conventional feedback controllers are no longer appropriate since industrial control systems are often decentralized (i.e., in-teractions among subsystems are not considered). Due to dynamic coupling it is well known that performance may be poor, and stability properties may be even lost. MPC provides several form of distributed approaches that guarantee nominal closed-loop stability and convergence to the centralized optimal performance. This thesis shows recent research activities on Dis-tributed MPC to demonstrate how is getting a mature technology, suitable to be applied to different application areas and large-scale systems, with computational as well as organizational advantages. This could lead to use MPC beyond its control aspects, in order to exploit its management capabil-ities. This work shows how Distributed Model Predictive Control, not only seems to fit the new technologies that are entering the global market, by creating a new interesting opportunities, but also could become a keyword for the emerging "smart factory".

(8)

(9)

SOMMARIO

M

PREDICTIVECONTROL (MPC) è una tecnica di controllo

avan-zato, ampiamente utilizzato nelle industrie di processo e ha le sua basi teoriche nel controllo ottimo. Inoltre, diverse tipologie di controllori predittivi sono stati sviluppati negli ultimi due decenni. Nono-stante ciò, i controllori con retroazione convenzionali (ad es. PID) sono, tuttora lo standard per la maggior parte delle applicazioni industriali. Ciò è dovuto al fatto che al controllore non è necessario dare alcuna informazione sul modello del sistema da controllare. D’altra parte, nel caso di applica-zioni su larga scala, i controllori convenzionali non sono più appropriati a causa del fatto che i sistemi di controllo industriale sono spesso decentra-lizzati (cioè, le interazioni tra i sottosistemi non sono considerate). A causa di ciò, le prestazioni potrebbero degradare e il sistema potrebbe persino diventare instabile. La letteratua su MPC, invece, presenta diverse forme di approcci di tipo distribuito che garantiscono stabilità nominale in ciclo chiuso e convergenza alle prestazioni ottimali centralizzate. Questo lavoro ha lo scopo di mostrare i recenti avanzamenti sul tema di Distributed MPC per dimostrare quanto stia diventando una tecnologia matura per essere ap-plicata a sistemi su larga scala e quanto riesca ad adattarsi a diversi campi applicativi, con vantaggi sia computazionali che organizzativi. Questo por-ta a utilizzare MPC non solo per i suoi aspetti controllostici, ma anche per le sue capacità di gestione ad alto livello. Pertanto, Distributed Model Predictive Control, non solo sembra andare bene con le nuove tecnologie che stanno entrando nel mercato globale, creando così nuove interessanti opportunità, ma potrebbe anche diventare un punto fisso della emergente "smart factory".

(10)

(11)

LIST OF PUBLICATIONS

International Journals

1. Razzanelli M, Crisostomi E, Pallottino L, Pannocchia G. Distributed model predictive control for energy management in a network of mi-crogrids using the dual decomposition method. Optim Control Appl Meth. 2019;1–17.

2. Antonelli, G., Arrichiello, F., Caiti, A., Casalino, G., De Palma, D., Indiveri, G., Razzanelli, M., Pollini, L., Simetti, E. (2018). ISME activity on the use of Autonomous Surface and Underwater Vehicles for acoustic surveys at sea. ACTA IMEKO, 7(2), 24-31.

3. Razzanelli, M., Pannocchia, G. (2017). Parsimonious cooperative dis-tributed MPC algorithms for offset-free tracking. Journal of Process Control, 60, 1-13.

International Conferences/Workshops with Peer Review

1. Razzanelli, M., Innocenti, M., Pannocchia, G., and Pollini, L. (2019). Vision-based Model Predictive Control for Unmanned Aerial Vehi-cles Automatic Trajectory Generation and Tracking. In AIAA Scitech 2019 Forum (p. 1409).

2. Pollini, L., Razzanelli, M., Pinna, F., Indiveri, G., Simetti, E., Alibani, M., and Innocenti, M. (2018, October). Development of the Guidance Navigation and Control System of the Folaga AUV for Autonomous Acoustic Surveys in the WiMUST Project. In OCEANS 2018 MT-S/IEEE Charleston (pp. 1-6). IEEE.

(12)

3. Razzanelli, M., Pannocchia, G. (2016). Parsimonious cooperative distributed MPC for tracking piece-wise constant setpoints. IFAC-PapersOnLine, 49(7), 520-525.

4. Pollini, L., Razzanelli, M., Olivari, M., Brandimarti, A., Maimeri, M., Pazzaglia, P., Pittiglio, G., Nuti, R., Bülthoff, H. (2016). Design, realization and experimental evaluation of a haptic stick for shared control studies. IFAC-PapersOnLine, 49(19), 78-83.

Book Chapters

1. Razzanelli, M., Aringhieri, S., Franzini, G., Avanzini, G., Giulietti, F., Innocenti, M., Pollini, L. (2016, June). A Visual-Haptic Display for Human and Autonomous Systems Integration. In International Work-shop on Modelling and Simulation for Autonomous Systems (pp. 64-80). Springer, Cham.

2. Franzini, G., Aringhieri, S., Fabbri, T., Razzanelli, M., Pollini, L., Innocenti, M. (2016, June). Human-Machine Interface for Multi-agent Systems Management Using the Descriptor Function Frame-work. In International Workshop on Modelling and Simulation for Autonomous Systems (pp. 25-39). Springer, Cham.

3. Pollini, L., Antonelli, G., Arrichiello, F., Caiti, A., Casalino, G., De Palma, D., Indiveri, G., Razzanelli, M., Simetti, E. (accepted for publi-cation). AUV Navigation, Guidance and Control for Geoseismic Data Acquisition In F. Ehlers (Ed.), Autonomous Underwater Vehicles -Design and Practice. Stevenage, UK: IET.

(13)

NOTATIONAL CONVENTIONS

Symbols, operators and functions

Notation Description

A 0 (A < 0) Positive (semi-)definite matrix A.

A> _{Transpose of matrix A.}

A−1 _{Inverse of matrix A.}

rank(A) Rank of matrix A.

ker(A) Kernel of matrix A.

Ai,j Entry of matrix A on i-th row and j-th column.

A ⊕ B set addition of sets A and B.

A B set difference of sets A and B.

diag(ai), i ∈ {1, . . . , n} Diagonal matrix with entries aion its diagonal. In n × n identity matrix - denoted as I if the

dimen-sion is clear from the context.

0n×m n × m zero matrix - denoted as 0 if the dimension is clear from the context.

||x||P Weighted 2-norm

√

x>_{P x of a vector x ∈ R}n and a matrix P ∈ Rn×n.

||x|| 2-norm of vector x ∈ Rn.

0 _{denotes an optimal cost or vector.}

diag{T1. . . TM} or diag (T1. . . TM)

represent block diagonal concatenation, of the supplied matrices {T1. . . TM}.

hor{T1. . . TM} or hor (T1. . . TM) represent horizontal concatenation, of the sup-plied matrices {T1. . . TM}.

col(A) represents the column-wise vectorization opera-tor applied to matrix A.

(14)

Abbreviations and acronyms

Abbreviation Meaning

MPC Model Predictive Control.

D-MPC Distributed MPC.

CD-MPC Cooperative D-MPC.

LTI Linear-Time Invariant.

DP Dynamic Programming.

FHOCP Finite Horizon Optimal Control Problem.

DNLTI Discrete Non-LTI.

CLF Control-Lyapunov Function.

GCLF Global CLF.

AS Asymptotic Stability.

ES Exponential Stability.

GAS Global AS.

GES Global ES.

QP Quadratic Program.

SSTO Steady-State Target Optimizer.

MG Micro-Grid.

EMS Energy Management System.

DER Distributed Energy Resource.

DNO Distributed Network Operator.

LP Linear Program.

ESS Energy Storage System.

UAV Unmanned Aerial Vehicle.

VS Visual Servoing.

IBVS Image-Based Visual Servoing.

PBVS Position-Based Visual Servoing.

NED Noth-East Down.

DCM Direction Cosine Matrix.

GNC Guidance, Navigation and Control.

NMPC Non-Linear MPC.

(15)

Sets

Notation Description

R Set of real numbers.

N Set of natural numbers.

Z Set of integer numbers.

B Binary set.

Rn Set of real vectors with n components.

Rn×m Set of real matrices with n rows and m columns. R>0 (R≥0) Set of positive (nonnegative) real numbers.

(16)

(17)

Part I

(22)

(23)

CHAPTER

1

AN INTRODUCTION TO MODEL

PREDICTIVE CONTROL

1.1

Introduction and outline

Model Predictive Control (MPC) algorithms were born in industrial en-vironments (mostly refining companies) during the 70’s, [39, 110]. This was due to the necessity to satisfy the more stringent production requests (e.g., economic optimization, maximum exploitation of production capac-ities, minimum variability in product qualities). Nowadays, most complex plants especially in refining and (petro)chemical industries use MPC sys-tems. After an initial reluctance, the academia “embraced” MPC contribut-ing to establish theoretical foundations and developed new algorithms.

The basic idea of using MPC is to consider a dynamic model of the real system to control. Then, by collecting measurements from sensors, MPC is capable to compute the best control action (in terms of minimization/maxi-mization of the objective to achieve) in order to produce a desirable system behavior. At each time interval, only the first control action of the

(24)

com-puted optimal input sequence is applied. Thus, modeling is a central task in MPC, and it is also its drawback. Since the optimal control move depends on the initial state of the dynamic system, and in general we have no direct access to it, a second basic concept in MPC is to measure the output of the dynamic system to estimate the current state of the system.

Figure 1.1 summarizes how MPC works.

Time t Past reconciliation Sensors estimate data prediction Input seq. Future prediction Actuators

Figure 1.1: Model Predictive Control framework: at each time t the controller retrieve the measurements from sensors and compute the optimal control move (i.e., to apply to the actuators) in order to obtain a suitable future behavior of the system.

Both the regulation problem, in which a model forecast is used to pro-duce the optimal control action, and the estimation problem, in which the past record of measurements is used to produce an optimal state estimate, involve dynamic models and optimization. The latter is performed by pre-dicting the system behavior in the future time instant or steps. The predic-tion is obtained by applying at each predicted step the corresponding con-trol action found by the optimization algorithm. As we detail next, MPC is able to ensure closed-loop stability, constraint satisfaction and robustness for multivariable linear and nonlinear systems [61, 77].

(25)

1.1. Introduction and outline

Despite the fact, MPC is widely used in process industries since ’80s, several class of controllers were developed in the last three decades, [79]. During the preliminary research activity, a comprehensive literature analy-sis was conducted in order to highlight the different MPC approaches de-veloped. Figure 1.2 summarizes this evolution.

Figure 1.2: Model Predictive Control in history.

This plot depicts briefly the evolution of MPC, in which each high-lighted time instant corresponds to an year in which a relevant research paper was published in the corresponding area (e.g., Linear [80], Non-Linear [100], Hybrid [21], Explicit [23], Robust [22,36], Adaptive [6], Eco-nomic [106], Fast [24], Stochastic [20], Distributed [119], Pursuit-Evasion [69], Passivity-based [104], Certified [7], Plug and play [112], Parsimo-nious [109], Scalable [47]).

Despite this preliminary but undoubtedly impressive informal introduc-tion on MPC, convenintroduc-tional feedback controllers (e.g., PID) are, so far, the standard de-facto for most industrial applications. This is due to the fact, es-sentially, that no system model is necessary for the standard controllers and the tuning procedure is quite intuitive provided from the fact that control actions depend only from the tracking error. However, no direct optimiza-tion is possible by using standard control approach as well as the fulfillment of, eventually hard, constraints on the input and output variables.

Further, in large-scale applications, it is useful (sometimes necessary) to have distributed or decentralized control schemes (rather than a unique centralized controller), where local control inputs are computed using local measurements, [117, 121]. Although details on predictive control for large-scale systems will be explained next, it is useful to recall that conventional feedback controller users, in case of large-scale systems, prefers a decen-tralized control structure in which a single (e.g. PID) controller for each

(26)

single input-output variable pairing occurs. In these cases, no assumptions on stability of the overall plantwide system can be guaranteed. As a mat-ter of fact, "trial and error" is the only available tuning strategy. On the other hand, MPC provides several kind of distributed approaches that try to establish feasibility as well as stability and optimality conditions.

Further, the rapid development of the Information and Communication Technology has stimulated the proliferation of distributed MPC algorithms due to the increasing size of the problems that can be addressed. The trend to consider control of larger systems, led to evaluate how MPC can be com-pliant also in the key societal infrastructures [89], such as urban traffic [33], water network [91], electricity grid [58], logistical networks [87], supply chains [43, 88], Internet of Things (IoT) [64] and Cyber Physical System (CPS) [124]. A recent research paper, [90], presents an overview and a roadmap on the use of MPC for large-scale systems, in which some future research and industrial opportunities are proposed. These applications have been gathered in Figure 1.3, that shows in which fields MPC became popu-lar through the history and a prediction on potential near future scenario in which MPC could become the most used control technique in few years.

Figure 1.3: Application of MPC in large-scale systems in history.

Even if it is quite difficult to carry out an extensive investigation to ob-tain a coherent overview, it is intuitive to state that today there is an

(27)

op-1.1. Introduction and outline

portunity: advanced control techniques (e.g. MPC) could have a direct impact on the society as a whole by using them beyond their control as-pects, by exploiting their management capabilities. In particular, we want to demonstrate how MPC framework can be suitable to be applied in dif-ferent control fields and large-scale systems. In order to accomplish this task, this thesis aims at analyzing the problem of controlling large inter-connections of subsystems in different application areas. Advantages and limits of (Distributed) Model Predictive Control are discussed in the first part of the thesis. Then, a first contribution in Distributed MPC theory is presented to deal with large-scale interconnected systems. Then, a second part is devoted to apply Distributed MPC in different control fields to de-velop innovative and advanced solutions. In the final part, all the relevant results are summed up. Then, motivated by the fact that new technologies such as the Internet of Things and Industry 4.0 essentially concern large interconnections of subsystems, a final section tries at providing some di-rections and advice for future research activities.

Finally, the thesis is organized as follows. The main purpose of Chapter 1 is to provide a compact and accessible overview of the essential elements of Model Predictive Control theory. Chapter 2 concerns the fundamen-tals of Distributed MPC approach and a comparison among different non-centralized control methodologies. Based on these latter results, a recent research topic in Distributed Model Predictive Control is addressed, and a novel control approach is proposed and analyzed in Chapter 3. Then, from the application standpoint, a Guidance, Navigation and Control approach based on a comparable control strategy with respect to that one of Chapter 3, is exploited in Chapter 4. An optimal power flow scheduling based on Distributed MPC is then explored in Chapter 5 to assess the capabilities of MPC to go beyond the control aspect. Finally, in Chapter 6 a review of the obtained results is shown and an outlook of the future of the Distributed MPC applications that could yields fruitful results are mentioned.

(28)

1.2

A predictive control formulation

The main goal of this section is to provide the principal tools of predictive control by analyzing a special form of MPC. It is very useful to understand how MPC works and which are the basic conceptual blocks that compose the MPC functionalities. We start with an Linear-Time Invariant (LTI) un-constrained system to cope the aforementioned task.

1.2.1 Linear Quadratic problem

Let us first consider an LTI system. We design a controller to steer the state of the deterministic, linear system to the origin. The system model is

x+= A x + B u

y = C x (1.1)

By using the model (1.1) we can predict how the state evolves given any set of inputs. Consider N time steps into the future and collect the computed input sequence into

u = {u(0), u(1), . . . , u(N − 1)}

We first define an objective function V (·) to measure the deviation of the trajectory of x(k), u(k) from zero by summing the weighted squares

V (x(0), u) = 1 2 N −1 X k=0 (x(k)TQx(k) + u(k)TRu(k)) + 1 2x(N ) T Pfx(N ) (1.2) subject to the constraint represented by the state dynamic evolution in Equa-tion(1.1). The objective function depends on the input and state sequence. Since, in this special case we do not consider a measurement noise and we assume C = I, the initial state at each time instant is available. The state trajectory, x(k), k = 1, . . . , N , is determined by the model and the input sequence u. Thus, the objective function shows explicit dependence on the input sequence and the initial state. The tuning parameters in the controller are the matrices Q and R. We allow the final state penalty to have a

(29)

differ-1.2. A predictive control formulation

ent weighting matrix, Pf , for generality. Large values of Q in comparison

to R reflect the designer’s intent to drive the state to the origin quickly at the expense of large control action. Penalizing the control action through large values of R relative to Q is the way to reduce the control action and slow down the rate at which the state approaches the origin. Choosing ap-propriate values of Q and R is not always obvious, and MPC inherits this tuning challenge. From the Equation (1.2) we can formulate the following optimal LQ control problem

min

u V (x (0) , u) (1.3)

The Q, Pf and R matrices often are chosen to be diagonal, but we do not

assume that here. We assume, however, that Q, Pf , and R are real and

symmetric; Q 0 and Pf 0 are positive semidefinite; and R 0

is positive definite. These assumptions guarantee that the solution to the optimal control problem exists and is unique.

Optimizing multistage function

A method to solve optimization problem as (1.3) is an iterative strategy called dynamic programming that we briefly exposed. Suppose the follow-ing three variables, (x, y, z), problem to be optimized

f (w, x) + g (x, y) + h (y, z)

Notice that the objective function has a special structure in which each stage’s cost function in the sum depends only on adjacent variable pairs. For the first version of this problem, we consider w to be a fixed parameter, and we would like to solve the problem

min

(30)

where w is fixed. we can obtain the solution by optimizing a sequence of three single variable problems defined as follows

min x f (w, x) + min y h g (x, y) + min z h (y, z) i

The iterative strategy solves the inner problem over z, then it moves to the next optimization problem and solve for y variable, and the final optimiza-tion is over x. This nested soluoptimiza-tion approach is an example of a class of techniques known as Dynamic Programming (DP) [25]. The version of the method we just used is called backward DP because we find the variables in reverse order: first z, then y, and finally x. Notice we find the optimal solutions as functions of the variables to be optimized at the next stage. Backward DP is the method of choice for the regulator problem.

Dynamic programming solution

After this brief introduction to DP, we apply it to solve the LQ problem (1.3). We first rewrite the equation (1.3) such that:

V (x(0), u) =

N −1

X

k=0

` (x(k), u(k)) + `N(x(N ))

where the stage cost ` (x, u) = 1₂ xT_{Qx + u}T_{Ru and the terminal stage}

cost `N(x) = 1₂ xTPfx. We assume the initial state x(0) is given. We

first rearrange the objective function to highlight the backward DP features by optimizing over input u(N − 1) and state x(N )

min u(0),x(1),...,u(N −2),x(N −1) ` (x(k), u(k)) + . . . + (1.4a) min u(N −1),x(N )` (x(N − 1), u(N − 1)) + `N(x(N )) (1.4b) subject to x(k + 1) = A x(k) + B u(k)

(31)

1.2. A predictive control formulation

The problem to be solved is the equation (1.4b) subject to

x(N ) = A x(N − 1) + B u(N − 1)

in which x(N − 1) appears in this stage as a parameter. We denote the optimal cost by V_{N −1}0 (x(N − 1)) and the optimal decision variables by u0_{N −1}(x(N − 1)) and x0_N(x(N − 1)). The optimal cost and decisions at the last stage are parameterized by the state at the previous stage as we expect in backward DP.

It can be shown that, given this form of the cost function, the optimal input for u(N −1) is K(N −1)x(N −1), in which K(N −1) = −(BTPfB+

R)−1BTPfA, defining the optimal control law at stage N − 1 to be a linear

function of the state x(N − 1). Then using the model equation, the optimal final state is also a linear function of state x(N − 1). The optimal cost is

1

2x(N − 1)

T

(K(N − 1)TRK(N − 1)

+ (A + BK(N − 1))TPf(A + BK(N − 1)))x(N − 1)

which makes the optimal cost a quadratic function of x(N − 1). Summa-rizing, for all x

u0_{N −1}(x) =K(N − 1)x x0_N(x) =(A + BK(N − 1))x V_{N −1}0 =1 2x T Π(N − 1)x Π(N − 1) =Q + ATPfA + K(N − 1)T(BTPfB + R)K(N − 1)+ 2K(N − 1)TBTPfA =Q + ATPfA − ATPfB(BTPfB + R)−1BTPfA

The function V_{N −1}0 (x) defines the optimal cost to go from state x for the last stage under the optimal control law u0_{N −1}(x). Having this function allows us to move to the next stage of the DP recursion. Notice that this problem is identical in structure to the stage we just solved, (1.4b), and we can write out the solution by simply renaming the variables. The recursion

(32)

from Π(N − 1) to Π(N − 2) is known as a backward Riccati iteration. To summarize, the backward Riccati iteration is defined as follows

Π(k − 1) = Q + ATΠ(k)A − ATΠ(k)B(BTΠ(k)B + R)−1BTΠ(k)A

for k = N, N − 1, . . . , 1 with terminal condition Π(N ) = Pf. The

termi-nal condition replaces the typical initial condition because the iteration is running backward. The optimal control at each stage is

u0k(x) = K(k)(x), k = N − 1, N − 2, . . . , 0

The optimal gain at time k is computed from the Riccati matrix at time k +1 as KN = − B0Π(k + 1) B + R −1 B0Π(k + 1) A, k = N − 1, . . . , 0

and the optimal cost to go from time k to time N is

V_k0(x) = 1 2x

T_Π(k)x, _{k = N, N − 1, . . . , 0}

Infinite horizon LQ regulator

In general, it is not true that a system with optimal control law is necessarily stable, [62]. Assume that we use as our control law the first feedback gain of the finite horizon problem, K(0),

u(k) = K(0)x(k)

Then the stability of the closed-loop system is determined by the eigenval-ues of A + BK(0). Let us consider the example as reported in [61]:

A = " ₄ 3 − 2 3 1 0 # B = " 1 0 # C = h −2 3 1 i

We define Q = CT_{C + 0.001 · I, in order to force Q positive definite and}

(33)

N = 5, we obtain

eig(A + BK5(0)) = {1.307, 0.001}

The closed-loop system is unstable. If we continue iterating the Riccati equation, we converge to the following steady-state closed-loop eigenval-ues

eig(A + BK∞(0)) = {0.664, 0.001}

The infinite horizon regulator asymptotically stabilizes the origin for the closed-loop system. Define the infinite horizon objective function

V (x, u) = 1 2 ∞ X k=0 x(k)TQx(k) + u(k)TRu(k) (1.5)

subject to (1.1) with Q, R > 0. Let us recall the Hautus Lemma for con-trollability:

Lemma 1.1. A system is controllable if and only if

rank [λI − A, B] = n _{∀λ ∈ C} (1.6)

in which C is the set of complex numbers.

If (A, B) is controllable in terms of (1.6), the solution to the optimiza-tion problem (1.3) exists and is unique for all x.

We denote the optimal solution with u0(x) and the first input in the optimal sequence with u0(0; x) = u0_{(x) (i.e., the feedback control law for}

the infinite horizon case, (1.5)).

Lemma 1.2. For (A,B) controllable, the infinite horizon LQR with Q, R 0 gives a convergent closed-loop system

x+ = Ax + Bu0(x) (1.7)

This means that the infinite horizon linear quadratic regulator (LQR) is stabilizing. From previous section the optimal solution is found by iterat-ing the Riccati equation. The optimal infinite horizon control law and the

(34)

optimal cost are given by u0(x) = K x, V0(x) = 1 2x 0 Πx in which K = −B0Π B + R −1 B0Π A, (1.8a) Π = Q + A0Π A − A0Π B (B0Π B + R)−1B0Π A, Π(∞) = Π (1.8b)

Lemma 1.2 demonstrate that for (A, B) controllable and Q, R 0, a pos-itive definite solution to the discrete algebraic Riccati equation (DARE), (1.8b), exists and the eigenvalues of (A + BK) are asymptotically stable for the K corresponding to this solution, [25].

The aforementioned LQR problem can be considered as a special form of the MPC, i.e., without any other constraints with respect to the system dynamics model or uncertainties. However, we can enlarge the class of systems and penalties for which the closed-loop stability is guaranteed. The system model restriction can be weakened from linear to non-linear system as well as from unconstrained to constrained system. This concludes the remainders of the background for understanding the MPC theory we show next.

1.2.2 MPC framework: model, objective function and constraints The regulators as explained in Sections 1.2.1 and 1.2.1 can be considered a special, but useful, form of MPC. However, there is a more general form of MPC that stabilizes linear and even nonlinear systems. In order to insert the MPC in a more general framework, let us first point out the differences between standard feedback controllers and MPC. The basic idea of MPC is that the control action is obtained by solving online, at each sampling instant, a Finite Horizon Optimal Control Problem (FHOCP) in which the initial state is the current (eventually estimated) state of the plant. From the optimization procedure we obtain a finite control sequence, and the

(35)

first control action is applied to the plant. The conventional controllers computes the control law offline.

If the current state of the system being controlled is x, MPC obtains, by solving an open-loop optimal control problem for this initial state, a specific control action u to apply to the plant. Dynamic programming tech-nique, Section 1.2.1, may be used to solve a feedback version of the same optimal control problem, however, yielding a receding horizon control law for all the possible states. The important fact is that if x is the current state, the optimal control u obtained by MPC is a receding horizon optimal con-trol law for x. DP would appear to be preferable since it provides a concon-trol law that can be implemented simply (as a look-up table). Obtaining a DP solution is difficult, if the state dimension is high. The great advantage of MPC is that open-loop optimal control problems often can be solved rapidly enough, using standard mathematical programming algorithms, to permit the use of MPC even though the system being controlled is nonlin-ear, and hard constraints on states and controls must be satisfied. Moreover, these classical controllers action depends only on tracking error and they do not handle constraints (safety, quality, operation). MPC considers directly pairing among input and output variables and it optimizes certain costs by respecting constraints and adjusting all manipulated variables simultane-ously.

In this Section we study MPC for the case when the state is known and there is no uncertainties. This case is particularly important, even though it rarely arises in practice, because important properties, such as stability and performance, may be easily established. We refer to stability in the absence of uncertainty as nominal stability.

General Formulation

Most physical systems can be described in the form of nonlinear differential equations as,

dx

(36)

The optimal control law in terms of closed-loop properties is the solution to the following infinite horizon, constrained optimal control problem,

P(x) : min u(·) V∞(x, u (·)) (1.10) subject to ˙x = f (x, u) (1.11a) x (0) = x0 (1.11b) u (t) ∈ U (1.11c) x (t) ∈ X (1.11d)

for all t ∈ (0, ∞). The cost is defined to be

V∞(x, u (·)) =

Z ∞

0

l (x (t) , u (t)) dt (1.12)

where x (t) and u (t) satisfy (1.11a) and l (·) is the stage cost. If l (·) is positive definite, the goal of the regulator is to steer the state of the system to the origin. We denote the solution to this problem (when it exists) and the optimal value function as

u0_∞(·; x) V_∞0 (x)

respectively. The closed-loop system under this optimal control law evolves as dx (t) dt = f x (t) , u 0 ∞(t ; x) (1.13) If l (·) and f (·) are differentiable, certain growth assumptions are satisfied and there are no state constraints, a solution to (1.10) exists for all x; V0

∞(·)

is differentiable and satisfies ˙

(37)

Using the formulation above with upper and lower bounds on V∞0 (·)

en-ables global asymptotic stability of the origin to be established. Although the control law u0_∞(0; ·) provides excellent closed-loop properties, there are several problems. A feedback, rather than an open-loop, control is usu-ally necessary because of uncertainty. Solution of the problem yields the optimal control for the state x but does not provide a control law. Dynamic programming may, in principle, be employed, but is generally impractical if the state dimension and the horizon are not small. In this section we re-strict the problem problem (1.10) with a more easily computed approximate Discrete Non-Linear-Time Invariant (DNLTI) problem. We also replace the semi-infinite time interval with a finite time interval and append a terminal region such that we can approximate the cost to go once the system enters the terminal region.

We develop here MPC for the control of constrained nonlinear time-invariant systems. The nonlinear system is described by the nonlinear dif-ference equation

x+ = f (x, u) (1.14)

where x ∈ Rn _{is the current state, u ∈ R}m _{is the current control and x}+

is the successor state. Function f (·) is assumed to be continuous and to satisfy f (0, 0) = 0, i.e. 0 is an equilibrium point. If the initial state is x (0) and the input is u(·), any solution of (1.14) satisfies

x (k + 1) = f (x (k) , u (k)) k = 0, 1, . . .

The solution of (1.14) at each instant k if the initial state at time 0 is x and the control sequence is u, is denoted by φ(k; x, u) that depends only by control sequence u = {u(0), u(1), . . . , u(N − 1)}. Because the system is time invariant, the solution does not depend on the initial time.

Proposition 1.3. Suppose the function f (·) is continuous. Then for each k ∈ I, function (x, u) → φ(k; x, u) is continuous.

The system (1.14) is subject to hard constraints which may take the form

(38)

where set I denotes the set of integers, I≥0 := 0, 1, . . ..

After the system dynamics mode, the next ingredient of the optimal con-trol problem is the cost function. Practical considerations require that the cost be defined over the finite horizon N to ensure the resultant optimal control problem can be solved sufficiently rapidly to permit effective con-trol. We consider initially the regulation problem where the target state is the origin. Since (1.14) is time invariant, the stage cost are all time irrele-vant and since the initial time is irreleirrele-vant, we define the FHOCP as PN(x).

Similarly, we indicate u0(x) and x0_{(x) as the best control action for state x}

and the optimal state sequence, respectively.

The optimal control problem PN(x) may be expressed as the

minimiza-tion of

N −1

X

k=0

` (x(k), u(k) + Vf (x(N )))

with respect to the decision variables (x, u) subject to (1.11b), (1.14) and (1.15). Note that u and x denote the control and state sequence

u = {u(0), u(1), . . . , u(N − 1)}

x = {x(0), x(1), . . . , x(N )} (1.16) Consequently, the optimal input and state sequence for the current state x are defined as,

u0(x) = {u0(0; x), u0(1; x), . . . , u0(N − 1; x)}

x0(x) = {x0(0; x), x0(1; x), . . . , x0(N ; x)} (1.17) For the purpose of analysis is preferable to constrain the state sequence x a priori to be a solution of x+ = f (x, u) enabling us to express the problem in the equivalent form of minimizing, with respect to the decision variable u, a cost that is purely a function of the initial state x and the control sequence u. This formulation is possible since the state sequence x may be expressed, via the difference equation x+ _{= f (x, u), as a function of (x, u). The cost}

(39)

1.2. A predictive control formulation becomes defined by VN(x, u) = N −1 X k=0 `(x(k), u(k)) + Vf(x(N ))

where, now, x(k) := φ(k; x, u) for all k ∈ I0:N. Similarly the hard

con-straints (1.15), together with an additional terminal constraint

x(N ) ∈ Xf (1.18)

where Xf ⊆ X impose an implicit constraint on the control sequence of the

form

u ∈ UN(x) (1.19)

in which the control constraint set UN(x) is the set of control sequences

u := {u(0), u(1), . . . , u(N − 1)} satisfying the state and control con-straints. It is therefore defined by

UN(x) := {u|(x, u) ∈ ZN}

in which the set ZN ⊂ Rn× RN mis defined by

ZN := {(x, u)|u(k) ∈ U, φ(k; x, u) ∈ X, ∀k ∈ I0:N −1, φ(N ; x, u) ∈ Xf}

Hence, the optimal control problem PN(x) may be expressed as

PN(x) : VN0(x) := min

u {VN(x, u)|u ∈ UN(x)} (1.20)

Problem PN(x) is a parametric optimization in which the decision variable

is u, and both the cost and the constraint sets depend on the parameter x. The set ZN is the set of admissible (x, u), i.e., the set of (x, u) for which

x ∈ X and the constraints of PN(x) are satisfied. Let XN is the set of states

in X for which PN(x) has a solution

(40)

It follows from (1.20) and (1.21) that

XN = {x ∈ Rn|∃u ∈ RN mwith(x, u) ∈ ZN}

It can be proved that not every optimization problem has a solution. We have to prove the existence of the solution that will be useful in the following. Let us assume,

Assumption 1.4. Continuity of system and cost. The function f : X×U → Rn, ` : X × U → R≥0 and Vf : Xf → R≥0are continuous and f (0, 0) = 0,

`(0, 0) = 0 and Vf(0) = 0.

Assumption 1.5. The sets X and Xf are closed, Xf ⊆ X and U are

com-pact; each set contains the origin.

The sets U, X and Xf are assumed to contain the origin because the first

problem we tackle is regulation to the origin. This assumption is modified when we consider the tracking problem.

Proposition 1.6. Existence of solution to optimal control problem. Sup-pose Assumptions 1.4 and 1.5 hold. Then

(i) The function VN(·) is continuous in ZN;

(ii) For each x ∈ XN, the control constraint set UN(x) is compact;

(iii) For each x ∈ XN, a solution to PN(x) exists.

For each x ∈ XN, the solution (1.20) is

u0(x) = arg min

u

{VN(x, u)|u ∈ UN(x)} (1.22)

In MPC, the control applied to the plant is the first element of u0(x), i.e., u0(0; x). At the next sampling instant, the procedure is repeated for the successor state. Although MPC computes u0(x) only for a specific value of x, may be used for every x for which PN(x) is admissible, obtaining the

implicit state feedback control law defined by u0(0; x), where x ∈ XN.

Continuity of the value function V0

N(·) and the implicit control law

u0_{(0; x) are desirable but it is not true in general. However, for a few}

(41)

1.3. Stability of MPC

Theorem 1.7. Continuity of value function and control law. Suppose that Assumptions 1.4 and 1.5 hold.

(i) Suppose there is no state constraints such that X = Xf = Rn. Then

the value function V0

N : XN → R is continuous and XN = Rn;

(ii) Suppose f (·) is linear, i.e., x+ = A x + B u, and that the state X and control U constraints sets are polyhedral (i.e., they can be represented as a system of linear inequalities). Then function V_N0 : XN → R is

continuous;

(iii) If, in addiction, the optimal control problem solution u0_{(x) is unique}

at each x ∈ XN, then the implicit MPC control law u0(0; ·) is

contin-uous.

The proof is detailed in [61].

1.3

Stability of MPC

1.3.1 Stabilizing Condition

In general, when constraints are present, or the system is non-linear, a Global CLF is seldom available. Hence, we should employ as our terminal cost function Vf a local CLF, one that is defined only on a neighborhood

Xf of the origin where Xf ⊆ X. A consequent requirement is that the

ter-minal state must be constrained, explicitly or implicitly, to lie in Xf. The

stabilizing condition is then defined as, Assumption 1.8. Basic stability assumption.

min

u∈U{Vf(f (x, u)) + `(x, u)|f (x, u) ∈ Xf} ≤ Vf(x), ∀ x ∈ Xf (1.23)

This assumption implies the following other one.

Assumption 1.9. Implied invariance assumption. The set Xf is control

invariant for the system x+ _{= f (x, u), i.e., there exists u ∈ U such that}

(42)

Assumptions 1.8 and 1.9, specify properties which, if possessed by the terminal cost function and terminal constraint set, enable us to employ the value function V_N0 for the optimal control problem PN as a Lyapunov

func-tion. However, also the following two properties (descent and monotonicity properties) are necessary for the value function V0

N(·).

Lemma 1.10. Optimal cost decrease. Suppose that Assumptions 1.4 and 1.5 hold and that Assumptions 1.8 and 1.9 hold. Then,

V_N0(f (x, u0(0; x))) ≤ V_N0(x) − `(x, u0(0; x)) ∀x ∈ XN (1.24)

Lemma 1.11. Monotonicity of the value function. Suppose that Assump-tions 1.4 and 1.5 hold and that AssumpAssump-tions 1.8 and 1.9 hold. Then

V_j+10 (x) ≤ V_j0(x) ∀x ∈ XN, ∀ j ∈ I0:N −1 (1.25a)

V_N0(x) ≤ Vf(x) ∀ x ∈ Xf (1.25b)

To proceed, we postulate two alternative conditions on the stage cost l (·) and terminal cost Vf(·) required to show that VN0(·) has the desired

properties such that stability holds:

Assumption 1.12. Bounds on stage and terminal costs; two condi-tions are possible:

(i) the stage cost `(·) and terminal cost Vf(·) satisfy

`(x, u) ≥ α1(|x|) ∀ x ∈ XN, ∀ u ∈ U (1.26a)

Vf(x) ≤ α2(|x|) ∀ x ∈ Xf (1.26b)

in which α1(·) e α2(·) are functions K∞;

(ii) the stage cost `(·) and the terminal cost Vf(·) satisfy

`(x, u) ≥ c1|x|a ∀ x ∈ XN, ∀ u ∈ U (1.27a)

Vf(x) ≤ c2|x|a ∀ x ∈ Xf (1.27b)

(43)

With this extra Assumption V_N0(·) has the following properties on the optimal value function.

Proposition 1.13. On the optimal value function properties.

(i) Suppose Assumptions 1.4, 1.5, 1.8, 1.9 and 1.12.(i) are satisfied. Then, there exist K∞ functions, α1(·) and α2(·) such that VN0(·) has

the following properties

V_N0(x) ≥ α1(|x|) ∀x ∈ X (1.28a)

V_N0(x) ≤ α2(|x|) ∀x ∈ Xf (1.28b)

V_N0(f (x, u0(0; x))) ≤ V_N0(x) − α1(|x|) ∀x ∈ XN (1.28c)

(ii) Suppose Assumptions 1.4, 1.5, 1.8, 1.9 and 1.12.(ii) are satisfied. Then, there exist positive constants c1, c2 and a such that VN0(·) has

the following properties

V_N0(x) ≥ c1|x|a ∀x ∈ X (1.29a)

V_N0(x) ≤ c2|x|a ∀x ∈ Xf (1.29b)

V_N0(f (x, u0(0; x))) ≤ V_N0(x) − c1|x|a ∀x ∈ XN (1.29c)

These properties are almost identical to those required in Theorems A.4 and A.5 to establish asymptotic stability of the origin with a region of at-traction XN. The problem is that the (1.28b), (1.29b) hold ∀ x ∈ Xf and

not for the region of attraction XN. The next results means that a generic

set X (e.g., Xf) may be XN, if X is compact, or a sublevel set of VN0(·).

Proposition 1.14. Extension of upper bound to compact set. Suppose that Assumptions 1.4, 1.5, 1.8 hold, that Xf contains the origin in its interior,

and that Xf ⊆ X where X is a compact set in Rn. If there exists a K∞

function α(·) such that V_N0_{(x) ≤ α(|x|)∀x ∈ X}f, then, there exists another

K∞function β(·) such that V_N0(x) ≤ β(|x|)∀x ∈ X.

A consequence of 1.13 and 1.14 is that:

Proposition 1.15. Suppose Assumptions 1.4, 1.5, 1.8, 1.9 and 1.12 are sat-isfied, that Xf has an interior containing the origin, and that XN is bounded.

(44)

Table 1.1: Stability Assumptions path to prove that Lyapunov theory can be used to estab-lish asymptotic stability of MPC in the general constrained case.

Assumption Imply Description or Proposition statement

1.4 - Continuity of system f and costs `(·), Vf(·).

1.5 - X, Xfclosed; U compact.

1.4+1.5 1.6 and 1.7 Existence of a solution and continuity of the value function V_N0.

1.8 1.9 Xf is control invariant for x+= f (x, u). 1.4+1.5+1.8 1.10 and 1.11 Optimal cost decrease and monotonicity of the

value function V_N0.

1.12 - Bounds on stage and terminal costs.

1.4+1.5+1.8+1.12 1.13 Optimal value function satisfy (A.2) except for (A.2b)

1.4+1.5+1.8 1.14 Holds if {0} ∈ Xf ⊆ X where X is a compact set ∈ Rn

1.16(+1.13+1.14) 1.15 Theorems A.4 and A.5 may be used to establish AS of the origin in XN.

Then for all x ∈ XN

V_N0(x) ≥ α1(|x|) (1.30a)

V_N0(x) ≤ α2(|x|) (1.30b)

V_N0(f (x, u0(0; x))) ≤ V_N0(x) − α1(|x|) (1.30c)

in which α1(·) and α2(·) are K∞functions.

Hence, Theorems A.4 and A.5 may be used to establish asymptotic sta-bility of the origin in XN. In the situations in which Xf has no origin in its

interior, we cannot establish an upper bound for V_N0(·) from Assumptions 1.8 and 1.9. Thus, we need the following assumption.

Assumption 1.16. Weak controllability. There exists a K∞function, α(·)

such that V_N0(x) ≤ α(|x|) ∀ x ∈ XN.

Assumption 1.16 relax the previous Assumption by highlighting those initial states that can be steered to Xf in N steps while satisfying the control

(45)

1.3.2 Summary

In the sequel we apply the previous results to establish asymptotic or expo-nential stability of a wide range of MPC systems. To facilitate application, we summarize these results and some of their consequences in the follow-ing theorem.

Theorem 1.17. On MPC stability

(a) Suppose that Assumptions 1.4, 1.5,1.8, 1.9, and 1.12.(i), are satisfied and that XN = Xf = Rn so that Vf(·) is a global CLF. Then the

origin is globally asymptotically stable for x+= f (x, u0_{(0; x)). If, in}

addition, Assumption 1.12.(ii) is satisfied, then the origin is globally exponentially stable.

(b) Suppose that Assumptions 1.4, 1.5,1.8, 1.9, and 1.12.(i), are satis-fied and that Xf contains the origin in its interior. Then the origin is

asymptotically stable with a region of attraction XN for the system

x+ _{= f (x, u}0_{(0; x)). If, in addition, Assumption 1.12.(ii) is}

satis-fied, and XN is bounded, then the origin is exponentially stable with

a region of attraction XN for the system x+ = f (x, u0(0; x)); if XN

is unbounded, the the origin is exponentially stable with a region of attraction that is any sub-level set of V_N=(·).

(c) Suppose that Assumptions 1.4, 1.5,1.8, 1.9, and 1.16 are satisfied and that `(·) satisfies `(x, u) ≥ α1(|x|) for all x ∈ XN, all u ∈ U, where

α1 is a K∞function. Then the origin is asymptotically stable with a

region of attraction XN for the system x+ = f (x, u0(0; x)). If `(·)

satisfies `(x, u) ≥ c1|x|a for all x ∈ XN, all u ∈ U and Assumption

1.16 is satisfied with α(r) = c2rafor some c1 > 0, c2 > 0 and a > 0,

then the origin is exponentially stable with a region of attraction XN

for the system x+_{= f (x, u}0_{(0; x)).}

(d) Suppose that Assumptions 1.4, 1.5,1.8 and 1.9 are satisfied, that `(·) satisfies `(x, u) ≥ c1|x|a+c1|u|aand Assumption 1.16 is satisfied with

α(r) = c2rafor some c1 > 0, c2 > 0 and a > 0. Then u0(0; x) ≤ c|x|

for all x ∈ XN where c =

c2

c1

1_a .

(46)

Table 1.2: MPC stability Theorems: synopsis.

Assumptions Imply Description

Theorem

1.4 + 1.5 + 1.8 + 1.12.(i) 1.17.(a) =⇒ GAS If XN _{= X}f = Rn so that Vf(·) is a GCLF.

1.4 + 1.5 + 1.8 + 1.12.(i) + 1.12.(ii)

1.17.(a) =⇒ GES If XN _{= X}f = Rn so that Vf(·) is a GCLF.

1.4 + 1.5 + 1.8 + 1.12.(i) 1.17.(b) =⇒ AS If Xf contains the origin in its interior. 1.4 + 1.5 + 1.8 + 1.12.(i) + 1.12.(ii) 1.17.(b) =⇒ ES If XN is bounded. 1.4 + 1.5 + 1.8 + 1.16 1.17.(c) =⇒ AS If `(x, u) ≥ α1(|x|)∀x ∈ XN, ∀u ∈ U, α1∈ K∞. 1.4 + 1.5 + 1.8 + 1.16 1.17.(c) =⇒ ES If `(x, u) ≥ c1|x|a_∀x _∈ XN_{, ∀u ∈ U, α(r) = c}2ra_. 1.4 + 1.5+1.8 + 1.16 1.17.(d), |u(x)| ≤ c|x| If `(x, u) ≥ c1|x|a_+c1|u|a_and

α(r) = c2ra_.

Proof of the Theorem 1.17 is detailed in [61].

We have not yet made any assumptions on controllability (stabilizabil-ity) or observability (detectabil(stabilizabil-ity). The reasons for this omission are that such assumptions are implicitly required, at least locally, for the basic sta-bility Assumption 1.8 and that we restrict attention to XN, the set of states

that can be steered to Xf in N steps satisfying all constraints. If the

sys-tem being controlled is linear, and the constraints polytopic or polyhedral, a common choice for Xf is the maximal invariant constraint admissible set

for a controlled system where the controller is linear and stabilizing. The terminal constraints set Xf is then the set {x|x(i) ∈ X, Kx(i) ∈ U} where

x(i) is the solution at time i of x+ _{= (A + BK)x and u = Kx is a}

stabi-lizing control law.

1.4

Example of MPC: LTI systems

We want to formulate an MPC example system to show the basic route to prove stability. Consider a Linear Time-Invariant system x+ _{= Ax + Bu}

(47)

1.4. Example of MPC: LTI systems

where X is closed and contains its origin) constraints. The stage cost is

`(x, u) = 1 2 x

T_{Qx + u}T_Ru

(1.31)

where Q and R positive definite. We know from Section 1.2.1, that if (A, B) is stabilizable, the solution to the infinite horizon unconstrained op-timal control problem Puc∞is known and the value function for this problem

is computed as

V_∞uc = 1 2x

T_{P x} _(1.32)

where P is the unique solution to the DARE

P = AT_KP AK + Q∗ (1.33)

in which AK = A + BK, Q∗ = Q + KTRK, u = Kx, and K is defined

by

K = − BTP B + R−1BTP AT (1.34) is the optimal controller. It is know that P is positive definite. We define the terminal cost

Vf(x) = V∞uc =

1 2x

T_{P x} _(1.35)

This terminal cost satisfies

Vf(AKx) +

1 2x

T_Q∗

x − Vf(x) ≤ 0 ∀x ∈ Rn

Given the current state x, we have to compute the optimal input sequence u0 _{by solving the following optimal control problem}

PN(x) : min

u VN(x, u) (1.36)

In order to ensure that u = Kx satisfies state and control constraints, the terminal constraints Xf must be a control invariant for x+ = AKx.

If these assumptions on Vf(·), Xf and `(·) hold, and Assumption 1.5

is satisfied, then Assumptions 1.8, 1.9 and 1.12.(ii) are satisfied, and Xf

contains the origin in its interior. Hence, by Theorem 1.17, the origin is asymptotically stable with region of attraction XN for the controlled system

(48)

x+ _{= Ax + Bu}0_{(0; x), and exponentially stable with a region of attraction}

any sub-level set of V_N0(·).

1.5

Suboptimal MPC

If the optimal control problem PN(x) solved online, is not convex, global

minimum of VN(x, u) cannot be determined in UN(x). Further, when

us-ing distributed MPC, it may be necessary or convenient to implement the control without solving the complete optimization. However, it is possible to achieve stability without requiring globally optimal solutions of PN(x).

All that is required is at state x, a feasible solution u ∈ UN(x) is found

giving a cost VN(x, u) lower than the cost VN(w, v) at the previous state

w due to the early control sequence v ∈ UN(w). The problem is analyzed

and solved in this single player scenario. Then, the same features arise in the distributed case.

Consider then the usual optimal control problem with the terminal cost Vf(·) and terminal constraint set Xf satisfying Assumptions 1.8 and 1.9;

X is assumed to be closed and U to be compact. In addition we assume that Vf(·) satisfies αf(|x|) ≤ Vf(x) ≤ γf(|x|)∀x ∈ Xf where γf(|x|)

and αf(|x|) are K∞ functions (that coincide with conditions (A.2a) and

(A.2b)). These conditions are satisfied if, for example, Vf(·) is a positive

definite quadratic function and Xf is a sub-level set of Vf(·) Let us also

as-sume Assumption 1.12.(i) satisfied. It is verified if `(·) is a positive definite quadratic function.

The basic idea behind the suboptimal model predictive controller is next considered. Let us assume the current state is x and that u ∈ UN(x). The

first element u (0) is applied to the system. Consider now the predicted control sequence u+_{defined by}

u = {u(1), u(2), . . . , u(N − 1), κf(x(N ))} (1.37)

where x(N ) = φ(N ; x, u) and κf satisfies Assumptions 1.8 for all x ∈ Xf.

Then, from Section 1.3.1, u+_{∈ U}

N satisfies

(49)

1.5. Suboptimal MPC

and, hence

VN(x+, u+) ≤ VN(x, u) − α1(|x|) (1.39)

From (1.39) we observe that no optimization is required for the cost re-duction (1.38). in practice u+ can be improved by several iterations of an optimization algorithm. Inequality (1.39) is similar to (A.2c). How-ever, applying the standard Lyapunov theory, is not possible due to the fact that there is no obvious Lyapunov function because at each x+ _{there exists}

many control sequence u+_{satisfying (1.39). Note also that the value}

func-tion VN(x, u) depends also from u+. However, global attractivity of the

origin in XN, may be established. The only modification required is when

x ∈ Xf such that,

VN(x, u) ≤ Vf(x) (1.40a)

f (x, u(0)) ∈ Xf (1.40b)

Stability of the origin can be established using (1.39), (1.40) and the prop-erties of Vf(·) as shown next in Section 1.5.2.

1.5.1 Suboptimal MPC algorithm

Let us consider a starting guess is available from the control trajectory at the previous time. The number of the steps of an optimization algorithm is fixed. Let us assume that each iteration is feasible and decreases the value of the cost function.

Step 1 Given a current state x = x0, we compute with respect to x0 a

feasible control sequence, u;

Step 2 Apply control u (0) to the system. Obtain state at next sample, x+ = f (x, u(0));

Step 3 Define the warm start as ˜u+= {u(1), u(2), . . . , u(N − 1), 0}; Step 4 The controllers performs Niter iterations of a feasible path

opti-mization algorithm to obtain an improved control sequence u+

(50)

Step 5 Update state and input sequence: x ← x+ and u ← u+. Go to Step 2 .

Note that in Step 3 , the warm start, is a simplified version of that one used in (1.37). In distributed MPC it is usual to choose zero for the final control move in the warm start.

1.5.2 Stability of Suboptimal MPC

It is possible to show that the system cost function VN(x, u) satisfies

cer-tain properties such that are similar to those required for a valid Lyapunov function, as described in Equation (A.2). The concerned property is that, it exists constants a, b, c > 0 such that:

a|(x, u)|2 ≤ V (x, u) ≤ v|(x, u)|2 V (x+, u+) − V (x, u) ≤ −c|(x, u(0))|2

The difference with respect to those described in Equation (A.2) is that the cost decrease depends only on x and the first element of u = u(0). Instead, in this case, it is sufficient to prove the convergence of x(k) and u(k) toward 0, but it can happen that u(k) → 0 while x(k) >>. This fact prevents us for establishing uniform Lyapunov stability nor Lyapunov stability for any k > 0 for the suboptimal value function VN(x, u). Let us now recall the

(uniform) Lyapunov stability definitions.

Definition 1.18. The zero solution x(k) = 0 for all k is stable (in the sense of Lyapunov) at k = k0 if for any ε > 0 there exists a δ(k0, ε)>0 such that

|x(k0)| < δ =⇒ |x(k)| < ε ∀k ≥ k0

Definition 1.19. The zero solution x(k) = 0∀k is uniformly stable (in the sense of Lyapunov) if for any ε > 0 there exists a δ(ε) > 0 such that

(51)

1.5. Suboptimal MPC

We add the following the constraint

|u| ≤ d|x| x ∈ rS (1.41)

in which d, r > 0, while S is ball in Rn of unit radius. This type of con-straint was first introduced in Equation (1.40). This provides (uniform) Lyapunov stability of the solution x(k) = 0 ∀k. The following lemma sum-marizes the conditions we use later for establishing exponential stability of Distributed MPC. A similar lemma establishing asymptotic stability of suboptimal MPC can be found in [120].

Definition 1.20. Let X be positive invariant for x+ _{= f (x). Then, the}

origin is exponentially stable for x+ _{= f (x) with a region of attraction X if}

there exists c > 0 and γ < 1 such that

|φ(i; x)| ≤ c|x|γi

for all i ≥ 0, x ∈ X.

Let us consider now the suboptimal MPC controller. The following re-lations hold

a|(x, u)|2 ≤ V (x, u) ≤ b|(x, u)|2 _{x ∈ X}

N, u ∈ UN

V (x+, u+) − V (x, u) ≤ −c|(x, u(0))|2 x ∈ XN, u ∈ UN

|u| ≤ d|x| x ∈ rS

The following Lemma holds.

Lemma 1.21. The origin is exponentially stable for the closed-loop sys-tem under suboptimal MPC with region of attraction XN if either of the

following assumptions holds

(a) U is compact. In this case, XN may be unbounded.

(52)

1.6

MPC for Tracking

It is a standard objective in applications to move the measured outputs of a system to a specified and constant setpoint, in general different from the origin, as explained in Section 1.2. This is the problem of the setpoint tracking. In this Section we consider first the case of tracking a constant reference signal for a non-linear and constrained system and we introduce the mathematical formalization. Then, we derive the same formulation for a linear and unconstrained system. Let us define the system to be controlled satisfies

x+= f (x, u) y = h(x) (1.42)

and is subject to the hard constraints

x ∈ X u ∈ U (1.43)

in which X is closed and U is compact. If the constant reference signal is r, then we want to steer the state x to a state ¯x such that y = h(¯x) = r. The target state and associated steady-state control are obtained by mini-mizing |¯u|2 with respect to (x, u) subject to the equality constraints (1.42) as well inequality constraints (1.43). Let us suppose that a solution (¯x, ¯u) exists. This means that the dimensions of r is less or equal to m, that is the dimension of u.

MPC then, solves online the following problem PN(x, r) defined by

V_N0(x, r) = min

u {VN(x, r, u)|u ∈ UN(x, r)} (1.44)

in which the cost function VN(·) is defined as

VN(x, r, u) = N −1

X

k=0

`(x(k) − ¯x(r), u(k) − ¯u(r)) + Vf(x, r)

and the constraints set is defined by

(53)

1.6. MPC for Tracking

Let u0(x, r) be the solution of (1.44). The terminal cost function Vf(·, r)

and constraint set Xf(r) must be chosen to satisfy suitably modified

stabi-lizing conditions. Since both depend on r, the simplest option is to choose a terminal equality constraint so that

Vf(¯x(r), r) = 0

Xf(r) = ¯x(r) ⊂ X

If the system is linear, an alternative choice is Vf(x, r) = Vf1(x − ¯x(r))

Xf(r) = ¯x(r) ⊕ X1f ⊂ X

in which V1

f and X1f are, respectively, the terminal cost function and

termi-nal constraint set derived in Section 1.4. In particular, V1

f will be equal to

that defined in Equation (1.35), i.e,

V_f1(x) = 1 2x

T_{P x}

1.6.1 Tracking for a LTI unconstrained system

Let us define the desired output as ytand denote a steady state of the system

model as (xs, us), i.e., h I − A −B i " xs us # = 0

For unconstrained system we can also impose, for now, that Cxs = yt for

the tracking problem. Thus, the following equilibrium relation must hold: " I − A −B C 0 # " xs us # = " 0 yt # (1.45)

If a solution exists, we can define the deviation variables as, ˜

x(k) = x(k) − xs

˜

(54)

that satisfy the dynamic model ˜

x(k + 1) = x(k + 1) − xs

= Ax(k) + Bu(k) − (Axs+ Bus)

= A˜x(k) + B ˜u(k)

(1.46)

so that the deviation variables satisfy the same model equation as the orig-inal variables. After solving the regulation problem in deviation variables, the input applied to the system is u(k) = ˜u(k) + us.

The matrix in Equation (1.45) is a (n + p) × (n + m) matrix. Thus, to have a solution for all yt, it is sufficient that the rows rows of the matrix

are linearly independent (i.e., p ≤ m). It is uncommon, in control appli-cations to have more measured outputs than manipulated inputs. However, to handle this problem, we choose a selection of linear combinations of the measured outputs, r = Hy ∈ Rnc_{, where n}

cis the number of the controlled

variable and we assign set-points to r. Thus, we define rt. Let us now

define the more general steady-state target problem, with respect to (1.45), as: min xs,us 1 2 |us− ut| 2 Rs+ |Cxs− yt| 2 Qs (1.47) subject to " I − A −B HC 0 # " xs us # = " 0 rt # Eus≤ e F Cxs≤ f where E = F = " I −I # e = " u u # e = " x x #

are chosen to simply define a lower and upper bounds for state and inputs. Let us make the following Assumptions on target feasibility and uniqueness

Assumption 1.22. On feasibility and uniqueness of the steady-state solution.

(55)

1.7. Offset-free MPC

(a) The target problem is feasible for the controlled variable setpoints of interest rt: a solution (x,us) exists.

(b) The steady-state input penalty Rs is positive definite: the solution is

unique.

Given the steady-state solution, we define the following dynamic regu-lation problem: min ˜ u V (˜x(0), ˜u) (1.48) subject to E ˜u ≤ e − Eus F C ˜x ≤ f − F Cxs

The objective function is defined as

V (˜x(0), ˜u) = 1 2 N −1 X k=0 |˜x(k)|2_Q+ |˜u(k)|2_R (1.49)

subject to (1.46). The optimal cost and the solution are V0_(˜_{x(0)) and}

˜

u0(˜x(0)). The moving horizon control uses the first input of the optimal control sequence by shifting it. Thus, the controller output will be

u(k) = ˜u0(˜x(0))

1.7

Offset-free MPC

Another common objective in applications is to use a feedback controller to compensate for an unmeasured disturbance to the system with the input in order to mitigate the disturbance’s effect on the controlled variable. The tracking error may converge to a nonzero constant vector, called the offset. Thus, the control is said to have the property of the zero offset. As in the previous Section, we first define the mathematical formulation in a non-linear system. Then we detail the non-linear case.

A simple method to compensate for an unmeasured disturbance is to model the disturbance. This choice is motivated by the works of [41, 51, 102, 122]. To achieve offset-free performance we augment the system state

(56)

with an integrating disturbance d. We assume the system to be controlled is

x+= f (x, u) d+= d

y = h(x) + d + v

in which v is the measurement noise. We assume the state is known, then we use a filter to obtain an estimate of ˆd. The filter is described as

ˆ

d+ = ˆd + L(y − h(x) − ˆd) (1.50)

From linear filtering theory we know that L must be chosen to ensure that I − L is stable. Since y − h(x) = d + v, the difference equation for ˆd may be written as

ˆ

d+= ˆd + L(d − ˆd + v)

We assume that a solution to this problem exists and we denote the solution by

¯

x(r, ˆd), ¯u(r, ˆd)_{. MPC may then be achieved by solving P}N(x, r, ˆd)

defined by V_N0(x, r, ˆd) = min u {VN(x, r, u)|u ∈ UN(x, r, ˆd)} in which VN(x, r, ˆd, u) = N −1 X k=0 `(x(k) − ¯x(r, ˆd), u(k) − ¯u(k)(r, ˆd)) + Vf(x, r, ˆd) and UN(x, r, ˆd) = {u|x(k) ∈ X, u(k) ∈ U, ∀k ∈ I0:N −1; x(N ) ∈ Xf(r, ˆd)}

The optimal input control sequence of the FHOCP results in u0_{(x, r, ˆ}_d).

If ˆd is constant, standard MPC theory shows, under suitable assumptions, that the constant target state ¯x(r, ˆd) is asymptotically stable for x+ ₌

f (x, u0(0; x, r, ˆd)) with a region of attraction XN(r, ˆd) = {x|UN(x, r, ˆd) 6=

(57)

1.7. Offset-free MPC

1.7.1 Offset-free MPC for a LTI system

Let us assume an integrating disturbance d driven by a white noise wdsuch

that

d+ = d + wd (1.51)

The augmented system model used is " x d #+ = " A Bd 0 I # " x d # + " B 0 # u + w y =h C Cd i " x d # + v (1.52)

where Bdand Cdare tuning parameters for estimating the disturbance. The

only restriction is that the augmented model must be detectable.

Lemma 1.23. The augmented system (1.52) is detectable if and only if the non augmented system (A, C) is detectable, and the following condition holds: rank " I − A −Bd C Cd # = n + nd (1.53)

Corollary 1.24. The maximal dimension of the disturbance d in (1.52), such that the augmented system is detectable is equal to the number of measurements, i.e., nd≤ p

We can now estimate ˆx(k) and ˆd(k) at each time k. The best forecast of the steady-state disturbance using (1.51) is to consider the disturbance equal to its mean value, that is ˆds = ˆd(k). The steady-state target problem

is modified as, min xs,us 1 2 |us− ut|2Rs + |Cxs+ Cd ˆ ds− yt|2Qs (1.54)