Extended Infinite Horizon MPC

We begin by introducing the Extended Infinite Horizon Model Predictive Control in the Formulation and results section, which is based on the Output Prediction-Oriented Model. The OPOM is a discrete time state-space representation that includes the input increments , the outputs y, and the states x_s and x_d. Subsequently, we present the cost function V of the controller, which contains an infinite-horizon term, along with a terminal state constraint on x_s that includes a slack variable .

We assume that the open-loop system described by the OPOM is stable (Assumption 1). Open-loop stability means that the open-loop system converges to any steady state when an input value is applied. It is relevant in practice because if the system is unstable, the EIHMPC cannot be implemented. Under this assumption, and by decomposing the infinite horizon cost V into two components, it is possible to reduce the infinite term to a finite horizon term with a terminal weight . Then, the formulation of the corresponding control optimization problem is presented. It includes both the terminal state constraint on x_s and the input constraints. At each time step k, the optimization variables are the sequence of input increments and the slack variable .

Next, we introduce Assumption 2, which states that the output reference (also called set-point) r is reachable from the input set . Here, reachability refers to the capacity of the system to reach r from some input within . Its relevance relies on the fact that if r is unreachable, the output cannot converge to that reference value. In practice, when r is unreachable, the input saturates (i.e., reaches the boundary of ) trying, but not succeeding, to bring the output to the set-point.

Based on Assumptions 1 and 2, Theorems 1 and 2 are established, which address the convergence and stability of the closed-loop system.

Formulation and results

The EIHMPC considers the following OPOM, which is a discrete time state-space model obtained from the step response [4,5]. For discrete time , let

(1)

and

(2)

where , , , , , , , and I is the identity matrix of dimension n_y. It is worth noting that the matrix D_d considered here corresponds to the matrix D^dFN in [4]. In the state equation (1), x_s is called the static part of the state of the system, while x_d is called the dynamic part. The vector u represents the inputs of the model and y stands for the outputs. Matrix D₀ is called the static gain of the system. We will not go deeper into the structure of the matrices appearing in the model, because it is not relevant for the present work. The interested reader may consult the above references for more details.

In the infinite horizon MPC setting, we denote by the input horizon. Since, at each time step , an optimization problem is solved for u over the horizon time interval of size m, we introduce the following notation: for , we define as the j-th move of the input solution of the optimization problem (to be defined below) at time step k. For the other variables of the model, we will use a similar notation. It is important to point out that, at each time step k, only the first move of the input solution of the optimization problem, , is implemented in the system. As a consequence, at each time step , ,

and, for ,

The EIHMPC is based on the following cost function

where , is the set-point, is a slack vector, , and are positive definite. To prevent the above cost from being unbounded, as discussed in [4], the following constraint is imposed

(3)

We will work under the following assumption on the OPOM, which was already present in [4].

Assumption 1. The system is stable, that is, the spectral radius of F is strictly smaller than 1.

Using Assumption 1 and (3), we obtain, for ,

(4)

where

is positive semidefinite. Also, note that .

Now, in order to state the control optimization problem, we first fix and rectangles in containing the origin 0. Let us also define

Control Optimization problem: The control optimization problem of the EIHMPC can be stated as follows: at each time step , we determine that minimizes V_k subject to constraints (3),

Without loss of generality, we assume that, at time 0, the system is in the steady state , , . Since and are convex sets and R is positive definite, the solution of the above optimization problem is unique. Let , denote the solution at time step k, and let be the corresponding cost,

where , and are obtained from , (1) and (2).

We also need the following

Assumption 2. The output reference r is such that for some .

We finally state the results about convergence and stability for the EIHMPC. We emphasize that we do not suppose that the gain matrix D₀ is regular in the following theorems.

Theorem 1 (Convergence). Under Assumptions 1 and 2, we can choose the matrix S such that

Theorem 2 (Stability). Under Assumptions 1 and 2, the controller is stable, that is, for any , there exists such that implies that for all .

Proof of Theorem 1

From Proposition 1 and Lemma 3, we deduce that . Indeed, recalling that for all , observe that

as .

Finally, we will show that . To this end, we first observe that using (4), since , if then . Thus, assume that . This implies that and therefore there exists some subsequence such that for all , . Now consider . For n large enough using the strategy , for all and (we can check that it is indeed feasible) we obtain by Lemmas 2 and 3

where G is from (8). This gives us the desired contradiction. □

Proof of Theorem 2

Recall that the system is assumed to be initially in the steady state , , , and consider the following feasible strategy for the control optimization problem at time step k = 1,

which has an associated cost . Since the above strategy is not necessarily the optimal one, and using Lemma 1, we have that for all . But, for any ,

and, on the other hand, by the parallelogram law,

for some positive constant C₁. Thus we have that for some positive constant C₂, for all . This implies that, for any , there exists such that for all , if . □

Extended Infinite Horizon MPC with zone control

In this section, we present the EIHMPC with zone control. The Formulation and results section introduces the controller formulation, which is based on the OPOM described previously in the Extended Infinite Horizon MPC section. This MPC formulation has two key features: the first one is that the set-point r is treated as an optimization variable; therefore, we denote it here as y_sp. The second feature is that the controller receives external input targets u_des.

We begin by defining the cost function and the terminal constraints: a terminal state constraint on x_s, and an input constraint that links the control inputs to the external target u_des. At each time step k, the optimization variables of this MPC are the sequence of input increments , the output reference vector y_sp,k, the slack variables associated with the terminal state constraint , and those slacks related to the input constraint .

For this controller, we work under Assumption 1 (which for convenience will be renamed as Assumption 3 below) and Assumption 4, which states that the input target u_des is reachable, i.e., belongs to the input domain , and the corresponding predicted steady state belongs to the set-point domain . Assumption 4 guarantees the consistency between the input target and the set-point domain. In practice, without this assumption, the controller will not be able to guide, at the same time, the input to its target and the output to the set-point.

Based on these assumptions, Theorems 3 and 4 establish the convergence and stability of the closed-loop system under the EIHMPC with zone control.

Formulation and results

In this section, we consider the EIHMPC with zone control [5]. This controller uses the OPOM given by (1) and (2) for prediction. The input target is sent at each time step by the Real Time Optimization (RTO) layer to the MPC layer. Unlike [4], in this MPC the set-point is a variable of the optimization problem and is calculated at each time step. Here, the exact values of the set-point are not important, as long as they remain inside a range with specified limits. As in the Extended Infinite Horizon MPC section, for , we define as the j-th move of the input solution of the optimization problem (to be defined below) at time step k. For the other variables of the model, we will use a similar notation.

The EIHMPC with zone control is based on the following cost function: for ,

where and are slack vectors, and , , , and are positive definite weighting matrices. To prevent the cost from being unbounded, as discussed in [5], we impose the terminal constraints

(10)

and

(11)

In this section we also work under Assumption 1, which for convenience we rename as

Assumption 3. The system is stable, that is, the spectral radius of the matrix F is strictly smaller than 1.

Under Assumption 3 and constraints (10) and (11) we have that, for ,

or, in other words,

where

is positive semidefinite and satisfies .

In order to state the control optimization problem, we first fix and rectangles in and rectangle in , all of them containing the corresponding origin. Let us recall that

Control optimization problem: The control optimization problem of the EIHMPC with zone control is the following: at each time step , we determine that minimizes subject to constraints (10), (11),

Here we assume that, at time 0, the system is in the steady state u(0) = 0, x_s(0) = 0, x_d(0) = 0. Since , and are convex sets and R and S_y are positive definite, the solution of the above optimization problem is unique. Let , , , be the solution of the control optimization problem at time step k and let be the corresponding cost,

where is obtained from , and , and are obtained from , (1) and (2).

In the following, we also need the

Assumption 4. The input target u_des is such that and .

We finally state our results about convergence and stability of this second controller.

Theorem 3 (Convergence). Let be the identity matrix of dimension n_u and H be the matrix given by

Under Assumptions 3 and 4, if then

Theorem 4 (Stability). Under Assumptions 3 and 4, the controller is stable, that is, for any , there exists such that implies that

Proofs of Theorems 3 and 4

The proof of Theorem 3 follows a similar reasoning to that of Theorem 1, although it is simplified by the presence of the target u_des for the input variable u. Indeed, while in Lemma 3 we only have that

now the presence of the input target u_des allows us to prove that

see Lemma 5. As a consequence, we obtain the apparently stronger fact that

see Corollary 1. This crucial fact enables us to prove by contradiction that as , for a well chosen S_u. The proof of Theorem 4 parallels the proof of Theorem 2.

We start with several technical results that will be useful to prove Theorems 3 and 4. The following lemma was established in [5], but for the sake of completeness we present its proof.

Lemma 4. Under Assumption 3, the sequence is non-increasing.

Proof. Let , , , be the following strategy for the control optimization problem at time step k + 1:

We can check that the above strategy is feasible for the control optimization problem at time step k + 1, and its cost is

Now, since and for , we have that

But observe that and then

On the other hand, we have that , so that

Moreover, note that and then . Thus, we deduce that

Finally, since the strategy , , , at time step k + 1 is not necessarily the optimal one, we have that and then we conclude that the sequence is non-increasing. □

Lemma 5. Under Assumption 3, we have that

Proof. From the proof of the last lemma, we have that

Since the sequence is non-increasing and non-negative, we deduce that it converges and then

(12)

Using (11), we have that

Thus, by (12), we obtain that

Now, observe that

Under Assumption 3, for any , there exists such that

and there exists such that, for all , . Finally, for all ,

that is, as . □

Lemma 6. Under Assumption 3, we have that

Proof. By Lemma 5, we can choose k large enough so that

For such a k, let , , , be such that

and observe that this is a feasible strategy for the control optimization problem at time step k. Indeed, we have that

and

so that (10) and (11) are satisfied, and we also have that

for . Moreover, note that and for .

Now, the cost corresponding to the above strategy is

and, for , we have that and

Moreover, we also have that

and then we obtain that

where

Hence, by Lemma 5, we deduce that

On the other hand, note that and , therefore

Finally, since the sequence converges, we conclude that

As a byproduct of the last lemma, we have the following □

Corollary 1. Under the same assumption of Lemma 6, we have that

Proof of Theorem 3

Assume that , where is the identity matrix of dimension n_u and

and suppose that .

By Corollary 1, we can choose a large enough k such that

(13)

For such a k, let be such that

and consider the following strategy for the control optimization problem at time step k:

It is straightforward to check that this strategy satisfies (11) and, from the fact that the solution of the control optimization problem at time step k satisfies (10) (11), we deduce that

which can be used to check that the above strategy also satisfies (10). Moreover, we have that , for , and

Using the fact that , which follows from (11), we also have that

since both and . Also, note that since both and . Thus, the above strategy is indeed feasible at time step k, and the corresponding cost is

Hence, we have that

Now, if then by equivalence of the norms, we have that

(where is the euclidean norm) and therefore there exists c > 0 such that infinitely often. Since

(14)

and, for ,

(15)

and also

(16)

by Lemma 5 and Corollary 1, for , there exists k₀ (large enough) satisfying (13) such that

since . Finally, as , we conclude that , which is a contradiction.

On the other hand, if then, by Lemma 6, we have that

and using (14), (15), (16), Lemma 5 and Corollary 1 we deduce that, for , there exists k₁ (large enough) satisfying (13) such that

and then , which is a contradiction once again. □

Proof of Theorem 4

Consider the following strategy for the control optimization problem at time step k = 1,

Since the system is assumed to be initially in the steady state , , , the above strategy is indeed feasible and it has an associated cost given by

Moreover, since this strategy is not necessarily the optimal one, and using Lemma 4, we have that for all .

Now, for any , we have that

On the other hand,

for some positive constant C₁ and

for some positive constant C₂. Thus, we obtain

for some positive constant C₃, for all , which finally implies the result. □

Appendix

In this appendix, we first obtain an explicit expression for the constant C₃ that appears in the proof of Proposition 1. For a positive semidefinite matrix A, let us define

where is the euclidean norm. It is well known that is the spectral radius of A. Starting back from the first inequality of (9), we have that the term

and therefore we obtain that

(17)

where and Z are defined in the proof of Proposition 1.

Next, we give a recipe to compute the matrix S from Theorem 1. This is done in three steps:

1. From matrix D₀, compute the matrix (see the paragraph just above Proposition 1 for the definitions of K₁, K₂ and L). Observe that in the case D₀ regular, the expression of boils down to .
2. Next, compute (or obtain an upper bound for) C₃ using expression (17).
3. Finally, choose and set .

Now, in order to illustrate the above recipe, we present the following toy example. Consider a system with n_u = 3, , , u_r = (1/2,1/2,1/2) and

Also, consider the input horizon m = 2 and weighting matrices Q = I₂ and R = I₃, i.e., the identity matrices of dimensions 2 and 3, respectively.

From D₀, and u_r, we deduce the following:

1. Computation of Ŝ:

We first obtain the matrix M of in the orthonormal bases of and ImD₀ (see the paragraph just above Proposition 1):

The LQ-factorization M = LO, with lower triangular L and orthogonal O, is given by

We also deduce that the matrix K₁ is simply the two-dimensional identity, while K₂ is scalar null. Thus, we have that

2. Upper bound for C₃:

Now, we obtain the matrix

which is the solution to the Lyapunov equation . Next, we compute the matrix Z from (8),

The spectral radii of Z, and are respectively , and . Thus, we obtain that

3. Deduction of S:

Finally, we choose , which gives

List of symbols

D₀, D_d, F, : matrices of the OPOM (see (1) and (2));

x_s: static part of the state of the OPOM;

x_d: dynamic part of the state of the OPOM;

u: input variable of the OPOM;

: input increment variable of the OPOM;

y: output variable of the OPOM;

m: input horizon for both controllers;

V: cost function of the EIHMPC;

Q, S: weighting matrices of the EIHMPC;

: slack variable of the EIHMPC;

r: set-point of the EIHMPC;

: cost function of the EIHMPC with zone control;

Q_u, Q_y, S_u, S_y: weighting matrices of the EIHMPC with zone control;

, : slack variables of the EIHMPC with zone control;

u_des: input target of the EIHMPC with zone control;

y_sp: set-point of the EIHMPC with zone control;

R: input increment weighting matrix for both controllers;

: input domain for both controllers;

: input increment domain for both controllers;

: set-point domain for the EIHMPC with zone control.