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We consider a set of continuous algebraic Riccati equations with indefinite quadratic parts that arise in H￥ control problems. It is well known that the approach for solving such type of equations is proposed in the literature. Two matrix sequences are constructed. Three effective methods are described for computing the matrices of the second sequence, where each matrix is the stabilizing solution of the set of Riccati equations with definite quadratic parts. The acceleration modifications of the described methods are presented and applied. Computer realizations of the presented methods are numerically compared. In addition, a second iterative method is proposed. It constructs one matrix sequence which converges to the stabilizing solution to the given set of Riccati equations with indefinite quadratic parts. The convergence properties of the second method are commented. The iterative methods are numerically compared and investigated.

Recently the algebraic Riccati equations with indefinite quadratic part have been investigated intensively. The paper of Lanzon et al. [_{¥}-approach to simulate optimal solutions under a flexible choice of system parameters. Here, a continuous H_{¥}-approach to jump linear equations is studied and investigated.

More precisely, how to find the stabilizing solution of the coupled algebraic Riccati equations of the optimal control problem for jump linear systems with indefinite quadratic part:

is considered. In the above equations the matrix coefficients

where

The stabilizing solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilizing solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The main idea is to construct two matrix sequences such that the sum of corresponding matrices converges to the stabilizing solution of the set of Riccati Equation (1). Such approach is considered in [

Here we introduce the sufficient conditions for the existence of stabilizing solutions of the set of Riccati Equation (1). We will prove under these conditions some convergence properties of constructed matrix sequences in terms of perturbed Lyapunov matrix equations. In addition, we introduce a second iterative method where we construct one matrix sequence. We show that the second iterative method constructs a convergent matrix sequence. Moreover, if the sufficient conditions of the first approach are satisfied then the second iterative method converges.

The notation

We use notation

eigenvalues to

We denote

We will rewrite the function

where

Note that transition coefficients

We introduce the following perturbed Lyapunov operator

and will present the solvability of (1) through properties if the perturbed Lyapunov operator.

Proposition 1: [

1) The matrix

2) The perturbed Lyapunov operator

The above proposition presents a deterministic characterization of a stabilizing solution to set of Riccati Equation (1).

A matrix

Knowing the stabilizing solution

Dragan et al. [

Each matrix

where

However, it is not explained in [

In our investigation we present a few iterative methods for finding the stabilizing solution to (5). Convergence

properties of the matrix sequence

second aim of the paper is to provide a short numerically survey on iterative methods for computing the stabilizing solution to the given set of Riccati equations. Results from the numerical comparison are given on a family of numerical examples.

Lemma 1. For the map

i)

for any symmetric matrices

ii)

with

Proof. The statements of Lemma 1 are verified by direct manipulations. □

Lemma 2. Assume there exist positive definite symmetric matrices

Then

i) if

ii) if

Proof. Assume the index i is fixed. We have

Thus

In order to prove the statement 2) we derive:

Since the matrices

The lemma is proved.

In this section we are proving the some convergence properties of the matrix sequences

Theorem 1. Assume there exist symmetric matrices

i) The Lyapunov operator

ii)

iii) The Lyapunov operator

iv)

Proof. The algorithm begins with

Under the assumption the Lyapunov operator

Using Lemma 1 1) and the fact that

is asymptotically stable and

The Lyapunov operator

Since

Thus, following Lemma 2, 1) we conclude that

Thus, the properties 1), 2), 3) and 4) are true for

Combining iteration (5) with equality

we prove by induction the following for

(a_{k}): The Lyapunov operator

(b_{k}):

(g_{k}): The Lyapunov operator

(d_{k}):

We have seen the statements (a_{0}), (b_{0}), (g_{0}) and (d_{0})) are true. We assume the statements (a_{k}), (b_{k}), (g_{k}) and (d_{k}) are true for

We know

Following Lemma 2, 2) the operator _{r}) and (b_{r}) are proved.

We have to prove the operator _{r}). Moreover, _{r}) is true for

Further on, we have

is asymptotically stable by Lemma 2, 2) Using again Lemma 2, 1) we conclude

The theorem is proved. □

The problem is to find the stabilizing solution

The Riccati Iterative Method. We choose

with

It is well know that if the matrix pair

Based on Riccati iteration (11) we consider the improved modification given by:

with

The Lyapunov Iterative Method. We choose

with

We consider the Lyapunov iteration (13) as a special case of the Lyapunov iteration introduced and investigated by Ivanov [

where

Convergence properties of the matrix sequence defined by (14) are given with Theorem 2.1 [

Further on, we consider an alternative iteration process where one matrix sequence is constructed. This sequence converges to the stabilizing solution of the given set of Riccati equations. We are proving that this in-

troduced iteration is equivalent to the iteration loop (4)-(5). We substitute

Thus, we can construct the matrix sequence

The unknown matrix

We have considered two iterative methods for computing the matrix sequence

(15) and the Lyapunov iteration (14). In the begining we remark the LMI approach for finding the stabilizing solution to (5). Following similar investigations [

has a solution which is the stabilizing solution to (5).

We carry out experiments for solving a set of Riccati Equation (1). We construct two matrix sequences

The matrices

algebraic Lyapunov equations for (14) at each step. For this purpose the MATLAB procedure care is applied where the flops are

Our experiments are executed in MATLAB on a 2.20 GHz Intel(R) Core(TM) i7-4702MQ CPU computer. We use two variables tolR and tol for small positive numbers to control the accuracy of computations. We de-

note

We consider a family of examples in case

and

and

In our definitions the functions randn (p, k) and sprand (q, m, 0.3) return a p-by-k matrix of pseudorandom scalar values and a q-by-m sparse matrix respectively (for more information see the MATLAB description). The following transition probability matrix

is applied for all examples.

For our purpose we have executed hundred examples of each value of m for all tests. _{M}” and the average number of iterations for the second iterative process “It_{S}” needed for achieving the relative accuracy for all examples of each size. The column “CPU” presents the CPU time for executing the corresponding iterations. Results from experiments are given in

We have studied two iterative processes for finding the stabilizing solution to a set of continuous-time genera-

n | (4)-(5) with RI: (15) | (4)-(5) with LI: (14) | (4)-(5) with LMI: (16) | ||||||
---|---|---|---|---|---|---|---|---|---|

It_{M} | It_{S} | CPU | It_{M} | It_{S} | CPU | It_{M} | It_{S} | CPU | |

Test 1: m_{1} = 4 | |||||||||

7 | 3 | 12.2 | 3.9 s | 3 | 12.6 | 1.6 s | 3 | 19.8 | 16.7 s |

8 | 3 | 14.7 | 4.6 s | 3 | 13.7 | 1.6 s | 4 | 20.3 | 23.5 s |

9 | 3 | 16.5 | 5.6 s | 3 | 16.4 | 2.3 s | 5 | 21.9 | 35.7 s |

10 | 4 | 17.4 | 6.5 s | 4 | 18.9 | 2.8 s | 4 | 23.4 | 54.8 s |

11 | 4 | 22.7 | 9.8 s | 4 | 20.5 | 3.3 s | 4 | 26.3 | 84.2 s |

12 | 6 | 27.3 | 13.3 s | 5 | 26.8 | 4.6 s | 4 | 31.6 | 130.5 s |

Test 2: m_{1} = n | |||||||||

7 | 3 | 12.7 | 4.0 s | 3 | 12.5 | 1.3 s | 4 | 20.7 | 20.8 s |

8 | 4 | 13.2 | 4.4 s | 3 | 14.9 | 1.8 s | 3 | 22.0 | 28.0 s |

9 | 4 | 15.6 | 6.2 s | 4 | 16.2 | 2.1 s | 3 | 22.3 | 39.8 s |

10 | 4 | 17.7 | 7.8 s | 4 | 18.4 | 2.5 s | 3 | 26.2 | 66.0 s |

11 | 4 | 20.5 | 9.7 s | 5 | 21.0 | 3.0 s | 4 | 37.2 | 125.3 s |

12 | 4 | 23.3 | 11.5 s | 4 | 22.8 | 3.3 s | 4 | 36.9 | 163.2 s |

13 | 4 | 25.1 | 11.6 s | 4 | 25.6 | 4.0 s | 5 | 57.0 | 371.0 s |

14 | 4 | 28.6 | 15.3 s | 4 | 27.3 | 4.7 s | 4 | 73.8 | 636.5 s |

n | the max number of iteration steps | the average number of iteration steps | CPU time |
---|---|---|---|

Iteration (15) for m_{1} = 4 | |||

7 | 30 | 17.0 | 1.8 s |

8 | 38 | 18.6 | 2.1 s |

9 | 28 | 19.7 | 2.5 s |

10 | 58 | 23.2 | 3.1 s |

11 | 56 | 27.5 | 4.2 s |

12 | 61 | 31.6 | 5.2 s |

Iteration (15) for m_{1} = n | |||

7 | 24 | 16.1 | 1.8 s |

8 | 27 | 17.7 | 2.1 s |

9 | 30 | 18.8 | 2.6 s |

10 | 29 | 20.9 | 3.3 s |

11 | 40 | 24.3 | 4.0 s |

12 | 51 | 25.5 | 4.3 s |

13 | 40 | 27.9 | 4.5 s |

14 | 46 | 31.6 | 5.2 s |

lized Riccati Equation (1). We have made numerical experiments for computing this solution and we have compared the numerical results. In fact, it is a numerical survey on iterative methods for computing the stabilizing solution. We have compared the results from the experiments in regard of the number of iterations and CPU time for executing. Our numerical experiments confirm the effectiveness of proposed new method (15).

The application of all iterative methods shows that they achieve the same accuracy for different number of iterations. The executed examples have demonstrated that the two iterations “(4)-(5) with RI: (15)” and “(4)-(5) with LI: (14)” require very close average numbers of iterations (see the columns “It_{S}” for all tests). However, the CPU time is different for these iterations. In addition, by comparing iterations based on the solution, the linear matrix Lyapunov equations shows that iteration “(4)-(5) with LI: (14)” is slightly faster than the second iteration (15). This conclusion is indicated by numerical simulations. Based on the experiments, the main conclusion is that the Lyapunov iteration is faster than the Riccati iteration because these methods carry out the same number of iterations.

The present research paper was supported in a part by the EEA Scholarship Programme BG09 Project Grant D03-91 under the European Economic Area Financial Mechanism. This support is greatly appreciated.

IvanG. Ivanov,Ivelin G.Ivanov, (2015) H_{∞} Optimal Control Problems for Jump. Journal of Mathematical Finance,05,337-347. doi: 10.4236/jmf.2015.54029