Finding a solution to strict inequalities

> 0, (B.13)

where S(x) is indeed an invertible matrix.

B.3 Finding a solution to strict inequalities

LMI problems may be composed of positive-definite (strict) inequalities and positive semi-definite (non-strict) inequalities. If all the conditions like Jⁱ(x) ≥ 0 are substituted by conditions like Jⁱ(x) > 0, the result is an LMI problem with only positive-definite inequal-ities. If there is a feasible solution x_f to the LMI problem with inequalities like Jⁱ(x) > 0, there is definitely a feasible solution to the one with inequalities like Jⁱ(x) ≥ 0. However, the converse is not true. In some cases, after the substitution, xf cannot be found, de-spite the fact that there is a feasible solution xf for the original LMI problem with the inequalities Jⁱ(x) ≥ 0. This is due to the fact that inequalities like Jⁱ(x) ≥ 0 may include an implicit equality like diag(P, −P ) ≥ 0 which, in turn, may have an implicit equality of P = 0. Another possibility is that the LMI is turned out to be singular, shown by J(P ) = diag(P, 0) ≥ 0 even if P > 0. Therefore, substitution of semi-definite conditions with definite conditions will not result in a feasible solution.

LMIs with inequalities like Jⁱ(x) ≥ 0 can be equivalently transformed to LMI with inequalities like Jⁱ(x) > 0 by removing implicit inequalities or nonsingular terms. In this way, the LMI problem can be further simplified by omitting any constant null-spaces. A typical problem that may arise in the transformation is, as mentioned before, the infeasibil-ity of any solution to the resulting LMI problem. At the same time, the best optimization algorithms for LMI solvers cannot deal with most of the inequalities like Jⁱ(x) ≥ 0 in an LMI problem unless they are transformed to inequalities like Jⁱ(x) > 0 (see the next sec-tion). If all hidden equality constraints and nonsingular terms are recognized and excluded from an LMI, the solver can find a solution to the LMI problem with strict inequalities in a similar way to the original LMI problem. This section attempts to address this problem by explaining how to recognize implicit equalities and nonsingular terms in the LMI stabi-lization problems of Theorem 4.5 and Theorem 4.6 to ease the transformation to positive definite inequalities; and hence to find the feasible solution in a similar manner to that of the original LMI problem.

When the fuzzy state-space region Ω^x_q encompasses the origin, it can be deduced that V_q(0) = 0 in the second condition of all stability theorems stated (Sections 4.3 and 4.4).

Vq = ˜x^TPqx in the (4.18) also becomes V˜ q = x^TPqx, meaning that the terms πq and pq

are equal to zero. Therefore, πq = 0 and pq = 0 are the first implicit equalities that should be considered if the transformation to positive-definite inequalities is necessary.

Moreover, there are additional implicit inequalities in the parameters associated with regions Ωq, substituted by semi-definite inequalities Q^k(x) ≥ 0, where Q^k(x) is defined as (4.29). For clarification, we reformulate the second, third and fourth conditions of the

171 B.3 Finding a solution to strict inequalities coroner of the above matrices should be positive semi-definite. On the other hand, all the elements in the lower right corner should also be ”≤ 0” [216]. Therefore the LMI elements P_sq

k=1λ^k_qd^k_q = 0,P_sq

k=1υ_q^kd^k_q = 0 and P_sq,mi

k=1 λ^k_q,mid^k_q,mi = 0. A matrix variable with ”0” in the lower right corner cannot fulfill a positive-definite inequality [216]. The only way to solve the above inequalities by positive-definite conditions, is transforming the singular matrix to a non-singular one by putting all the matrix elements in the last row and column asP_sq

k=1λ^k_qc^k_q = 0,P_sq

k=1υ_q^kc^k_q = 0 and PqB(mi) +P_sq,mi

k=1 λ^k_q,mic^k_q,mi= 0. In this manner, there is a possibility to transform the LMI conditions in Theorem 4.5 and 4.6 to positive-definite inequalities.

There are two ways to assign P_s

k=1λ^kd^k = 0 by either λ^k = 0 or d^k = 0. λ^k = 0 implies that Q^k(x) ≥ 0 is a redundant condition, which obviously conflicts with its initial purpose. It also means that the other conditions in the LMI problem should be fulfilled in the entire fuzzy state space, which, as already emphasized, is quite conservative and leads to an infeasible solution. Therefore, for regions Ω^x_q defined by the quadratic forms (4.29) and encompassing the origin, parameters d^k = 0 should be selected. The same arguments is valid for the elementsP_s

k=1λ^kc^k= 0.

Similarly, for the fifth condition of the stability theorems, if region Ω^x_r encompasses the origin, then πr= 0 and pr= 0. Therefore the condition can be reformulated as

5.

k=1λ^k_q,rd^k_q,r. Similarly, the singular matrix can be transformed to a non-singular matrix in case of parameters c^k_q,r = 0 when both regions Ω^x_q and Ω^x_r encompass the origin. In brief, for regions Ω^x_q encompassing the origin, the local Lyapunov function can be defined as Vq(x) = x^TPqx by letting πq = 0 and pq = 0 and all regions can be defined as Q^k(x) = x^TZ^kx by letting c^k = 0 and d^k = 0, k ∈ Is. The same applies for region Ω^x_r with the corresponding parameters, if it encompasses the origin.

It may be the case, specially close to the switching manifold (defined by switch sets), that in terms of matrix inequalities, the evolution of a trajectory should be considered

Chapter B: Linear Matrix Inequalities 172

bidirectional between regions Ω^x_q and Ω^x_r at states described by the same parameters in Q^k(x) = 0. Therefore, an implicit equality for the local Lyapunov functions Vq(x) and Vr(x), measuring the energy in the regions Ω^x_q and Ω^x_r respectively, can be introduced by explicitly formulating the conditions Vq(x) ≤ Vr(x) and Vr(x) ≤ Vq(x). In this case, the

which should be fulfilled for all states. For the states fulfilling the quadratic equality Q^k(x) = 0, it is necessary that ˜Pr≤ ˜Pq and ˜Pq ≤ ˜Pr, meaning that ˜Pq = ˜Pr should hold close to the switching manifold (see Figure below).

x Figure B.1: (a) The vector field is bidirectional between Ω^x_q and Ω^x_r. The circle on the surface show the turning point of the vector fields. (b) Two regions are split into four to avoid an implicit equality like ˜Pq = ˜Pr.

For transforming to positive-definite conditions, there are two ways to remove these im-plicit equalities. The first is to further split the former partitions with respect to Λqr

in such a way that a solution trajectory passes through Ω^x_q to Ω^x_r or unidirectionally if otherwise. The main drawback of this method is the increasing computational burden as a result of the increasing number of unknown LMI variables. Moreover, splitting regions with respect to Λqr as explained above is not an effortless task.

The second method, which is more practical, is to let ˜Pq ≡ ˜Pr. Since this equality condition only have to occur on hyperplanes Q^k_q,r= 0, a less conservative condition is to allow the fifth condition in stability theorems should be substituted by Linear Matrix Equality (LME) conditions. Similarly, if another matrix like ˜Psshould fulfill the condition ˜Ps≡ ˜Pr