is a Lyapunov function (See Definition 2.2 of [12]), where ∆Vk(·) denotes the Lya- punov difference w.r.t. time k. Existence of such non-symmetric Lyapunov candi- date function with the corresponding non-symmetric stability domains for constrained control linear systems is already established in the literature (See [11, 12] and the references therein).
Remark 3.2 As the control law is computed based on the present state of the system and is implemented only for present time instance, even while solving for an infinite horizon problem, hereforth, let us considerxk≡xk|k and similarlyuk≡uk|k ∀k ≥0.
3.6 Solving the cross-coupled-AREs
Solution of the pair of cAREs to get the values of the Lyapunov function matrices,PkI and PkII would fetch the optimal (saddle point) solution of the Nash game approach to mixedH2/H∞ MPC defined in eqn.(3.5). Here an infinite horizon is adopted for the following reasons -(i) nominal stability is guaranteed, (ii) the resulting coupled non-symmetric Riccati equations from the LQ game problem would be algebraic, which can be solved atleast by some numerical means‡. Freiling et al. [58], has given the necessary condition(s) that need to be satisfied, so as to get the solution pair of the cAREs. A special matrix is constructed in [58] from the cAREs given in equations (3.25) and (3.26) and the closed loop system equation (3.22), whose invariant subspace spanning generalised vectors form the solution of the cAREs.
Assuming that the matrix A is regular, the equations (3.25) and (3.26) and the equation (3.22) can be written using index transformation as,
˜ xk
¯ pk
¯¯
pk
(q+ 1) =M
˜ xk
¯ pk
¯¯
pk
(q) (3.29)
‡ Solving the coupled non-symmetric difference Riccati equations from a finite horizon LQ game, even numerically, is intractable.
3.6. Solving the cross-coupled-AREs 64
where q:=N −k, N is some finite horizon in future w.r.t. the current time instant k, ˜xq =x(N + 1−q). Let us define S1 :=γ−2B1B1T,S2 :=B2R−1B2T, and
M =
A−1 A−1S1 A−1S2
Q1A−1 AT +Q1A−1S1 Q1A−1S2
Q2A−1 Q2A−1S2 AT +Q2A−1S1
(3.30)
Although the above construction is meant for finite horizon open-loop game, the asymptotic behaviour for an infinite horizon problem, resulting in cAREs, can be obtained as q→ ∞.
Theorem 3.1 [58] (i) If S(PI, PII) := span
I PI PII
⊂ R3n×n is an invariant
subspace of M with det(I +S1PI+S2PII) 6= 0 then
PI PII
is a solution of the cAREs (3.25) and (3.26).
(ii) If
PI PII
∈R2n×n is a solution of the cAREs, then S(PI, PII)⊂R3n×n is an invariant subspace of M. Moreover, the inverse in the closed-loop matrix, i.e., Φ, is the matrix of restriction of M to S(PI, PII) w.r.t. the basis defined by the columns of
I PI PII
.
Then the generalised eigenvectors of M give the solutions of the cAREs, such that;
Let span(v1,· · · , vn) be anM-invariant subspace such that det(X)6= 0 for
X Y1
Y2
:=
(v1,· · · , vn), then
PI PII
:=
Y1X−1 Y2X−1
is a solution of the cAREs if det(I + S1PI+S2PII)6= 0.
However, in [58] only necessary conditions are given but not the sufficient con- ditions. Nevertheless, the choice of the appropriate generalized eigenvector which
3.6. Solving the cross-coupled-AREs 65
forms the solution of the cAREs is also not clear enough. However, in [59] the suffi- cient conditions are given by assuming Q1 = 0 and for some Q0 ≥ 0 for continuous time system open-loop games. This makes the 3n×3n matrix M into a 4n×4n matrix. Nevertheless, solving the open-loop Nash game using the approach in [59]
involves fixing many parameter matrices, which can be a cumbersome trial and error process, and the result, however, thus finally obtained is a symmetric solution.
There doesn’t exist any solid analytical method for finding the solution of the cAREs, other than those given in [58, 82], which are not complete. However, bringing the cAREs into applicability at least by some means of numerical solvability is of more concern from a control engineer’s perspective than finding the exact solution(s).
Following propositions and the subsequent theorem will yield the numerical solution of the cAREs.
Let us denote the time-varying state feedback control law, with controller gain Kku =R−1B2TPkIΦk, asκk(xk) :={Kkuxk|uk=Kkuxk ∈ U}.
Proposition 3.1 (Input constraint) For the input constraint, uk ∈ U, if given in terms of the inequality bounds such that umin ≤ uk ≤ umax can be normalized to lie within the bound −u¯≤u˜k≤+¯ufor the entire horizon. Then the input constraint condition in the optimisation problem is given as, ku˜kk22 =kκk(xk)k22 ≤u¯2. The above input constraint condition is used in the optimisation problem while solving for the solution of the cAREs.
Proposition 3.2 For an optimal controller using two-player LQ Nash games for the system given by (3.1) results in a pair of cross coupled algebraic Riccati equations as given in (3.25) and (3.26), to achieve a stable closed loop system using the control law (3.23). The stable and locally unique solution set (PkI∗, PkII∗) for the cAREs is also achievable even if the cAREs satisfy
0 = ǫ1k+Q1−PkI+ATPkI[I+B2R−1B2TPkI −B1γ−2B1TPkII]−1A (3.31) 0 = ǫ2k+Q2−PkII +ATPkII[I +B2R−1B2TPkI−B1γ−2B1TPkII]−1A (3.32)
3.6. Solving the cross-coupled-AREs 66
for some solution set in the open-hyper balls of radius {ǫ1k, ǫ2k} ∈ Rn×n around and
containing the origin.
Theorem 3.2 For a system defined by Σ : (A, B1, B2), with initial condition x0, the two-player Nash game based mixed H2/H∞ MPC yields an (sub-)optimal, stable and locally unique solution, for the pair of cAREs,
0 =ǫ1k+Q1−PkI+ATPkI[I+B2R−1B2TPkI−B1γ−2B1TPkII]−1A (3.33) 0 =ǫ2k+Q2−PkII+ATPkII[I+B2R−1B2TPkI−B1γ−2B1TPkII]−1A (3.34) satisfying the input constraint condition,
(kκk(xk)k22− |u¯|2) +s2k = 0 (3.35) where sk ∈ R and along with the saddle point optimal condition bounds (assuming kQ1k2 >kQ2k2),
xTkPkIxk ≥βk (3.36)
xTkPkIIxk ≤βk (3.37)
if it exists, such that (PkI∗, PkII∗) ≻0, for some open-hyper balls of radius {ǫ1k, ǫ2k} ∈ Rn×n around and containing the origin, such that the closed loop system (3.22) is stable, using the control law (3.23), satisfying the input constraint, uk∈ U,∀k ≥0.
Proof Let PkI∗ and PkII∗ be the (sub-)optimal, stable and locally unique solution set of the cAREs, (3.33) and (3.34), satisfying the conditions (3.36) & (3.37) and uk ∈ U (i.e., satisfying equation (3.35)), with x0 ∈ X0, for some open hyper balls of radius {ǫ1k∗, ǫ2k∗}∈ Rn×n, respectively, in the linear space. Now assume that there exists another solution set, ˜PI∗k and ˜PII∗k , which again satisfies (3.33) and (3.34) for some other hyper balls of radius{˜ǫ1k∗,˜ǫ2k∗}∈ Rn×n, respectively, such that ǫjk∗ =
˜
ǫjk∗+δj ∈Rn×n and δj ≪˜ǫjk∗ j = 1,2.
Likewise, there may exist a different solution set for a different choice of δ(.) for every other choice of solution set. Therefore, as |δj| → 0, in the linear space the
3.6. Solving the cross-coupled-AREs 67
open hyper balls converges to the (sub-)optimal solution,
|δlimj|→0ǫj∗k = ˜ǫjk∗+δj = ˜ǫj∗k, j = 1,2.
which in turn gives PkI∗ = ˜PI∗k andPkII∗ = ˜PII∗k , thus giving an (sub-)optimal, stable
and locally unique solution set to the cAREs.
Considering that there exists an invariant set Ω and the input constraint set U for the system Σ, the following definitions are in order:
Definition 3.2 (Domain of Attraction/Performance) Given a feedback control law uk : Rn→Rm, let
X ,{xk∈Ω|uk ∈ U} (3.38)
be the domain of attraction/performance of the system.
Definition 3.3 (Initial condition set) The set of initial states x0 for which an admissible control policy exists is defined as
X0 :=
x0 ∈ X ⊂Rn
U 6=∅ (3.39)
Also note that, in the present case it is assumed that X0 ⊆ X
Corollary 3.1 An (sub-)optimal stabilizing controller (3.23), such thatuk ∈ U,∀k≥ 0, for any given x0 ∈ X0 and xk ∈ X,∀k >0, exists, for the given controller param- eter {γ, Qi, R}i=1,2, iff the locally unique, optimal matrix solution pair of the cAREs (PkI∗, PkII∗) are positive definite.
Note, that the non-symmetric coupled algebraic Riccati equations, in general, will have non-symmetric matrices as their solutions. However, by relaxing the equality of the cAREs and with the definition of positive definite matrix (c.f. Definition 3.1) solutions, the slack variables {ǫ1k, ǫ2k} ∈Rn×n will share the burden of the non- symmetricalness of the exact solution matrices. As one of the objectives of the present work is to find some suitable numerical means to bring the cAREs resulting from LQ Nash games into applicability from an engineer’s perspective rather than
3.6. Solving the cross-coupled-AREs 68
to look for an exact solution, this relaxation method is adopted in the present work.
Moreover, from the results of [59], although they are for continuous domain case, the results in the present case are also expected to be symmetric. However, it is a known fact that this relaxation technique will only give a sub-optimal solution. Due to the calculation of a new solution at every time instant in a receding horizon fashion, even though the solution is sub-optimal, the existence of a feasible stabilisable solution is sufficient to ensure stabilityi.e., feasibility implies stability, (c.f. [140]), when the sub-optimal solution is nearer to the optimal solution. Furthermore, as any non- singular symmetric matrix can be diagonalised, without losing the matrix property, in the present task of solving the cAREs, the solution matrices are forcefully taken as diagonal matrices. On the other hand, considering the solution matrices as diagonal matrices, the number of decision variables (entries of the matrix) reduces from n(n+1)2 to simply n for a n×n symmetric matrix. Thus this approach ultimately yields the result as desired i.e., performance of the closed loop both in terms of its transient response and stability. That is, by using a LQ-Nash game approach to mixedH2/H∞
MPC, the closed-loop Lyapunov stability with monotonically decreasing values of cost function is satisfied, if positive definite matrices exists by solving equations (3.33)-(3.37).
The algorithm used to solve the Nash game based mixed H2/H∞ MPC is given below.
Algorithm
1. Measure the system states xk
2. vec
Equation(3.25) Equation(3.26)
(k −R−1B2TPIΦkxkk22−u¯2) +s2k
1
2xTkPkIxk−βk βk− 12xTkPkIIxk
=vec(0)
3. Obtain the solution matrices (PI, PII) by minimising ǫ1k, ǫ2k, sk, βk.
3.6. Solving the cross-coupled-AREs 69
4. Compute the value ofuk from equation (3.23) and apply it to the process and model.
5. Goto step 1.
Remark 3.3 When proportional state weighing matrices are used i.e.,Q1 =αQ2, α >
0, the two algebraic Riccati equations in (3.25) and (3.26) reduce into a single Riccati equation (c.f. [1], [2]) resulting in a symmetric solution matrix.
Remark 3.4 The choice of the two state weighing matrices, which offers additional degree of freedom in the controller design, is a criterion that can be tuned according to the designer’s need on the closed-loop performance, such that for the given choice of {γ, Qi, R}i=1,2 the closed-loop system is stable. In general, with several numer- ical experiments, it is found that kQ1k2 >kQ2k2 gives smooth closed-loop transient response and kQ2k2 >kQ1k2 gives aggressive transient response.
In the present work, an open-loop Nash game is solved online, subject to point- wise constraints, at every time instant with the new process measurement and with the notion of feedback. Thus, when such an open-loop constrained optimal control problem is solvable at every time instance for the desired performance measures, the generally used MPC approach of approximating the infinite horizon problem as a finite horizon control problem along with the suitable terminal conditions to ensure stability and robustness, as described in Section 1.2.2, is not required. In the present work, the entire domain of performance acts as the terminal set of regular MPC schemes (c.f. Chapter 4). This relaxes the designer from the burden of enlarging the domain of attraction of the terminal constraint set associated with the local terminal controller and/or the computation of the optimal length of control horizon of the MPC design.