New solution method of linear static output feedback design problem for linear control systems

Linear Algebra and its Applications 504 (2016) 204–227

Contents lists available at ScienceDirect

Linear Algebra and its Applications www.elsevier.com/locate/laa

New solution method of linear static output feedback design problem for linear control systems Vasiliy Ye. Belozyorov Department of Applied Mathematics, Dnepropetrovsk National University, Gagarin’s avenue 72, 49050, Dnepropetrovsk, Ukraine

a r t i c l e

i n f o

Article history: Received 7 August 2015 Accepted 1 April 2016 Available online xxxx Submitted by V. Mehrmann MSC: 93B25 93B52 93B55 93C05

a b s t r a c t A new method of construction of the linear static output feedback for linear control systems is offered. The essence of this method consists in construction of an initial approximation of the feedback matrix such that a numerical sequence, which generated by arbitrary iterative method and beginning from this initial approximation, converges to an exact solution of the feedback design problem. Examples are given. © 2016 Elsevier Inc. All rights reserved.

Keywords: Autonomous linear control system Linear static output feedback Modal control problem Pole placement Characteristic polynomial Exterior degree Newton’s iterative method

1. Introduction Consider the linear autonomous control system whose state equation is ˙ x(t) = Ax(t) + Bu(t), x(t) ∈ Rn , u(t) ∈ Rm , E-mail address: [email protected]. http://dx.doi.org/10.1016/j.laa.2016.04.001 0024-3795/© 2016 Elsevier Inc. All rights reserved.

(1)

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

205

and an output equation has the form y(t) = Cx(t), y(t) ∈ Rp .

(2)

Here Rn , Rm , Rp are real linear spaces of vector-columns of dimensionalities n, m, p; x(t) = (x1 (t), . . . , xn (t))T , u(t) = (u1 (t), . . . , um (t))T , y(t) = (y1 (t), . . . , yp (t))T are vectors of states, inputs, and outputs; A : Rn → Rn , B : Rm → Rn , C : Rn → Rp are real linear maps of appropriate spaces. Introduce into system (1) a linear feedback under the law u(t) = Ky(t) = KCx(t),

(3)

where K : Rp → Rm is a real linear operator. Then we have the so-called closed-loop ˙ control system x(t) = (A + BKC)x(t), whose properties completely are determined by properties of the operator A + BKC. If the operator C is irreversible, then feedback (3) is called an output feedback; in opposite case feedback (3) is called a state feedback. Fix any bases in spaces Rn , Rm , and Rp ; then the triple of operators A, B, C will be represented in the chosen bases by matrices A ∈ Rn×n , B ∈ Rn×m , and C ∈ Rp×n . Further, we will adhere only to these denotations. Let In ∈ Rn×n be the identity mapping. Feedback design problem. It is necessary to construct a real matrix K ∈ Rm×p such that the characteristic polynomial of the matrix A + BKC would coincide with a desired real polynomial of degree n: det(λIn − A − BKC) ≡ λn + a1 (K)λn−1 + . . . + an (K) = λn + a1 λn−1 + . . . + an ; a1 , . . . , an ∈ R. For practical applications it is usually required that the polynomial λn + a1 λn−1 + . . . + an be stable. At first sight it seems that this problem can be solved with the help of the known Hurwitz Theorem for minors of the special matrix, which is constructed from the coefficients a1 (K), . . . , an (K). However, its application results in research of a nonlinear inequalities system with respect to the unknown elements of matrix K, solutions of which it is impossible to find for the matrix K of large sizes. The feedback design problem of linear control laws for linear control systems (there are other names of this problem: pole placement by static output feedback or modal control problem) was first formulated in the strict statement by R.E. Kalman in his lecture at the 1th IFAC World Congress in 1960. The concepts of the controllability and observability were base concepts for this statement, which were first also introduced by R.E. Kalman at the same 1th IFAC World Congress. Importance of the controllability and observability conditions exists in that these conditions are necessary for solvability of the feedback design problem.

206

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

In our opinion, the whole development period of the modal control theory for finitedimensional systems can be separated in four stages. The first stage began approximately in 1960 (from the moment of the strict statement of this problem by R.E. Kalman) and continued to 1967, when W.M. Wonham [29] proved the theorem that if the state feedback control is used, then the complete controllability is a necessary and sufficient condition for the solvability of modal control problem; article [29] is a starting point for a book [30], in which the author applied linear algebra tools to the solution of practical tasks of linear control theory. The second stage (approximately 1968–1975) is already characterized by research of the output feedback design problem (the vector of states is inaccessible to direct observation). Here, the most essential result was given by H. Kimura [10], who proved that if m + p > n, then the pole placement problem for the closed-loop control system almost always has a real solution. The majority of articles for this period differ by a linear approach, which was actively used. (The basic idea of this linear approach consists in that the control system in the state space is transformed so that coefficients of the characteristic polynomial of the closed-loop system (or even their part) linearly depended on the coefficients of matrix K.) The third development stage is a twelve-year period (approximately 1976–1987), which is characterized that L.R. Fletcher et al. [8,13] found the necessary and sufficient solvability conditions of the modal control problem in the case m + p > n and the following restriction: a spectrum of the closed-loop system contains only different eigenvalues. Improvement attempts of methods of the papers [8,13] were also undertaken later (see, U. Konigorski [11]). In this work under the condition m +p > n a new analytical approach to the pole placement problem is based upon parametric eigen-structure assignment of linear time-invariant multi-variable systems in combination with a special explicit formulation of the pole assignment equations. Note that in the papers [8,11,13] it was actually shown that if condition m + p > n is not fulfilled, then the feedback design problem becomes substantially nonlinear (the linear approach mentioned above in this case is not suitable) and for a solution of this problem it is necessary to apply completely other methods. The use of these methods began in 1981 when the mathematicians R.W. Brockett and C.I. Byrnes [5] first essentially applied exterior algebra tools to the modal control problem. Here the solvability conditions of the modal control problem for the case mp = n were found. Their paper was the exact beginning of the fourth stage, which had been used to the end of twentieth century. Among the large number of publications on applications of exterior algebra in the modal control theory, it should be noted the significance of X. Wang [26,27], who for general linear systems proved the solvability of the feedback design problem in the case mp ≥ n. However, in articles [5,26,27] any of recommendations for practical realization of these new ideas were not offered. (An attempt of the development constructive procedure of solution of the feedback design problem in the case mp = n was undertaken in [1].)

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

207

It should also be mentioned that there were other directions of research on the output feedback design problem. The methods of exterior algebra were applied by M.S. Ravi, J. Rosenthal et al. [16–18] for research of a linear dynamic feedback. The basic result, which here was derived, is its determination of minimum order of a dynamical compensator resolving the modal control problem for system (1), (2). Linear matrix inequalities (LMI) and the Lyapunov functions method for the search of the static and dynamic output feedbacks stabilizing system (1), (2) were applied in [6,19,21–23,25]. A linear output feedback with dynamic high gain for global regulation of a class of nonlinear systems was proposed by L. Praly and Z.P. Jiang [15]. Construction attempts of the stabilizing linear feed-back for bilinear control systems were done in [2,3]. For the control tasks of tracking, disturbance rejection, model matching, and decoupling and not necessarily proper plants I. Blumthaler and U. Oberst [4] derive necessary and sufficient conditions for the existence of proper stabilizing compensators with proper and stable closed-loop system behaviors, parametrize all such compensators as input/output behaviors and not only their transfer matrices and give new algorithms for their construction. Moreover, they solve the problem of pole placement or spectral assignability for the complete feedback behavior. Relating to the last achievements in the design feedback problem it should be said that a novel two-step strategy for static output-feedback controller design was proposed by F. Palacios-Quinonero et al. [14]. In the first step, an optimal state-feedback controller is obtained by means of the LMI formulation. In the second step, a transformation of the LMI variables is used to derive a suitable LMI formulation for the static output-feedback controller. As noted by V.L. Syrmos et al. [20] that “although many approaches and techniques exist to approach different versions of the problem, no efficient algorithmic solutions are available”. Nevertheless, it is necessary to admit that results obtained in paper [4] in some sense decide the feedback design problem. However, these results have a theoretical character and intended rather for specialists on mathematical system theory, than for engineers dealing with concrete constructing of linear regulators. Algorithms offered in [4] are difficult to realize in the spaces of large dimensions. In addition, in paper [4] (see Theorems 3.15 and 3.27) a dynamic output feedback (but not the static output feedback) design problem was actually considered. Therefore, in the present article a new numeral solution method of the static output feedback design problem is offered. It seems to us that results obtained in the present paper will be a useful addition to the known results represented in [4]. Taking into account all of the above, there is no pretense on the specific achievements in the feedback design problem solution. Therefore, the present work must be considered foremost as a new approach at the choice of an initial approximation for all iterative

208

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

procedures resolving the feedback design problem. The advantage of this technique has an impact that is significant especially in the case in which the condition m + p > n is not satisfied. 2. Bezout’s states column model For most general linear control systems the feedback design problem can be solved if matrices A and B are reduced to the special form obtaining the title Bezout’s states column model [9,12,24,28]. In future we will assume that system (1), (2) is completely controllable and observable. It means that   rank B, AB, . . . , An−1 B = n

(4)

and ⎞ C ⎟ ⎜ ⎜ CA ⎟ ⎟ = n. rank ⎜ .. ⎟ ⎜ . ⎠ ⎝ CAn−1 ⎛

(5)

Let m ≤ p and the matrices B and C are matrices of full rank. (In applied problems precisely this situation takes place.) Thus, we suppose that rank B = m, rank C = p.

(6)

Remark that the number n can always be uniquely represented as n = km + l = l(k + 1) + (m − l)k,

(7)

where k = [n/m] > 0 is an integer part of number n/m; n ≥ m > l ≥ 0, and l is integer. Assume that ν1 = ν2 = . . . = νl = k + 1, νl+1 = νl+2 = . . . = νm = k. Then ν1 + ν2 + . . . + νm = n. Let

 η · (η − 1) · . . . · (η − ξ + 1) η! η ≡ Bin(η, ξ) = = ξ!(η − ξ)! 1 · ... · ξ ξ be a binomial coefficient. Here η, ξ are integers, 0 ≤ ξ ≤ η, and 0! = 1! = 1. Denote by b1 , . . . , bm ∈ Rn the columns of the matrix B in selected bases of spaces n R and Rm .

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

209

We can construct matrices Vi ∈ Rn×n for the following aspect: Vi = (B, AB, . . . , Ak−1 B, Ak bs1 , . . . , Ak bsl ). Here 1 ≤ s1 < . . . < sl ≤ m; i = 1, . . . , r = Bin(m, l). Definition 2.1. A square (m × m) matrix G in each row and column of which there is exactly one element equal to 1 and remaining elements are equal to zero is called a permutation matrix. We assume that even for one i det Vi = 0. (It is evident that if l = 0, then V0 = (B, AB, . . . , Ak−1 B).) Then there is a permutation {t1 , . . . , tm } of elements of the set {1, . . . , m} such that vectors bt1 , . . . , Aν1 −1 bt1 , bt2 , . . . , Aν2 −1 bt2 , . . . , btm , . . . , Aνm −1 btm

(8)

are linearly independent. Thus, BG = (bt1 , . . . , btm ). (We remind that the pair (A, B) is completely controllable; in this case without loss of generality it is possible to consider that G = Im and BG = (b1 , . . . , bm ).) Example 2.1. In this simple example we show the method of constructing the matrices Vi ∈ Rn×n . Let n = 8 and m = 3. Then we have 8 = k ·3 +l. From here it follows that k = 2, l = 2, r = Bin(3, 2) = 3, and 1 ≤ s1 < s2 ≤ 3. Consequently, we have three pairs of the numbers (s1 , s2 ): (1, 2), (1, 3), and (2, 3). Thus, we obtain V1 = (B, AB, A2 b1 , A2 b2 ), V2 = (B, AB, A2 b1 , A2 b3 ), V3 = (B, AB, A2 b2 , A2 b3 ). (At least for one matrix Vj , we have det Vj = 0, j ∈ {1, 2, 3}.) Denote by V a nonsingular matrix columns of which are vectors (8) and denote by vi the row-vector being the solution of the following system of linear equations: vi V = (0...0  |...| 0...01   |...| 0...0 ); i = 1, . . . , m. ν1

νi

(9)

νm

Instead of the variables x(t) and y(t) in system (1), (2) we introduce new variables x(t) and y(t) under the formulas x(t) = Zx(t) and u(t) = Gu(t), where T Z T = (v1T , (v1 A)T , . . . , (v1 Aν1 −1 )T , . . . , vm , (vm A)T , . . . , (vm Aνm −1 )T )T ∈ Rn×n ,

G = (gij ) ∈ Rm×m is the permutation matrix.

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

210

Thus, in new bases of spaces Rn and Rm the matrices A and B take the forms [6,9,24]: ⎛

Ac = ZAZ −1

A11 ⎜ .. =⎝ . Am1

··· Aii ···

⎞ ⎛ B11 A1m .. ⎟ , B = ZBG = ⎜ .. ⎝ . . ⎠ c Amm 0

··· Bii ···

0 .. .

⎞ ⎟ ⎠ , (10)

Bmm

where ⎛ ⎜ ⎜ Aii = ⎜ ⎜ ⎝

0 .. . 0

1 .. . 0

−ai,ηi−1 +1

−ai,ηi−1 +2

··· .. . ··· ···

⎞ 0 .. ⎟ . ⎟ ⎟ ∈ Rνi ×νi ; ⎟ 1 ⎠ −ai,ηi

η0 = 0, η1 = ν1 , η2 = ν1 + ν2 , . . . , ηi = ν1 + . . . + νi ; ⎞ ⎛ 0 ··· 0 ⎜ .. .. ⎟ .. ⎜ . . . ⎟ ⎟ ∈ Rνi ×νj , i = j; Aij = ⎜ ⎟ ⎜ 0 ··· 0 ⎠ ⎝ −ai,δj−1 +1 · · · −ai,δj δ0 = 0, δ1 = ν1 , δ2 = ν1 + ν2 ..., δi = ν1 + . . . + νj ; ⎛ ⎞ 0 ⎜.⎟ ⎜ .. ⎟ νi ⎟ Bii = ⎜ ⎜ ⎟∈R . ⎝0⎠ 1 Here integer positive numbers νi accept two values: νi = k + 1, if i = 1, . . . , n − km and νi = k, if i = n − km + 1, . . . , m, where k is the integer part of number n/m; i, j = 1, . . . , m. It is clear that if the matrices A and B are reduced to form (10), then after the replacement of the base in space Rn the matrix C will pass in the matrix Cc = CZ −1 = (c1 , . . . , cn ),

(11)

where ci are p-dimensional column vectors; i = 1, . . . , n. Further, it is possible to consider that G = Im . Since eigenvalues of the matrix A + BKC of the original closed-loop system coincide with the eigenvalues of the matrix Z(A +BKC)Z −1 = ZAZ −1 +ZBKCZ −1 = Ac +Bc KCc of the transformed closed-loop system, we will work with matrices Ac , Bc , and Cc . In this case, the matrix K of control law (3) for the origin and transformed systems will be the same. 3. Equations for feedback design problem based on Bezout’s states column model Consider two matrices: (Im |K) ∈ Rm×(m+p) and the polynomial matrix

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

211

⎛  ⎞ ⎞ ⎛ ⎛ ⎞  α1,ν +1 ... α1,ν +ν ⎞ α11 ... α1ν1 1 0 1 1 2   1 1 ⎜  ⎜⎜ α ⎟ 0⎟ ⎜⎜ 21 ... α2ν1 ⎟ ⎜ ⎟ ⎜ α2,ν1 +2 ... α2,ν2 +ν2 1 ⎟ ⎜ ⎟ λ λ ⎜   ⎜⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ .. ⎟ · ⎜ .. ⎟ · ⎜ ⎜ ⎟ ⎜ ⎟ D(λ) = ⎜ . ⎟ ⎜ .. ⎟ ⎜ . . . . . . . . . . ⎟ ⎜ .. ⎟ ⎜⎜ . . . . . . ⎜⎜ ⎟ ⎝ . ⎠ ⎜ ⎟ ⎝ . ⎠ ⎝ αm,ν1 +1 ... αm,ν1 +ν2 0 ⎠  ⎝⎝ αm1 ... αmν1 0 ⎠ λν1  λν2  −c1 ... − cν1 0 −cν1 +1 ... − cν1 +ν2 0 ⎛ ⎞ ⎞  α1,ν +...+ν ⎛ ⎞ ... α1n 0 1 m−1 +1  1 ⎜ α ⎟ ⎟ ⎜ 2,ν1 +...+νm−1 +1 ... α2n 0 ⎟ ⎜ ⎟⎟ λ ⎜ ⎟ ⎟ ⎜ ⎟ (m+p)×m .. ⎟ · ⎜ ⎟⎟ × ... ⎜ , (12) . ⎟ ⎜ .. ⎟⎟ ∈ R(λ) ⎜ . . . . . . . . . . . ⎜ ⎟ ⎝ . ⎠⎟ ⎝ αm,ν1 +...+νm−1 +1 ... αmn 1 ⎠ ⎠  λ νm  −cν1 +...+νm−1 +1 ... − cn 0 ⎛⎛

where Im ∈ Rm×m is the identity mapping, λ is an independent variable and R(λ) is a set of all real polynomials of degrees no more than ν1 . Let H ∈ Rq×l be a real matrix. Assume that q ≤ l. Denote by ∧q H = (w1 , . . . , wr ) = (H1,...,q−1,q , H1,...,q−1,q+1 , . . . , Hq+1,q+2,...,l ) an exterior degree of the matrix H. It is a row-vector of dimension r = Bin(l, q) coordinates of which are minors of degree q of matrix H located in a lexicographic order. (This means that the coordinate Hi1 ,...,ik ,...,iq is located in the row ∧q H, earlier than coordinate Hj1 ,...,jk ,...,jq if i1 = j1 , . . . , ik−1 = jk−1 , ik < jk ; k ∈ {1, . . . , q}.) Similarly the exterior degree can be defined for the case q ≥ l. In this case it is necessary to replace q → l. Then ∧l H is a column-vector of dimension r. Denote by F a real (Bin(m + p, m) × (n + 1)) matrix the first column of which has the form (1 0 ... 0)T . Theorem 3.1. For matrices (10), (11) there exists a matrix F such that the identity ⎛

⎞ λn ⎜ . ⎟ ⎜ .. ⎟ m m m ⎟ det(λIn − Ac − Bc KCc ) ≡ ∧ (Im |K) · ∧ D(λ) ≡ ∧ (Im |K) · F · ⎜ ⎜ ⎟ ⎝ λ ⎠ 1 takes place. Proof. Let S = (sij ) ∈ Rn×n be any real matrix a structure of which coincides with the structure of the matrix Ac in (10). In the linear control theory it is well known the identity det(λIn − S) ≡ det M (λ) [6,9,22,30], where

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

212

⎛

M11 (λ) · · · ⎜ .. .. M (λ) = ⎝ . . Mm1 (λ) · · ·

⎞ M1m (λ) ⎟ .. m×m ; ⎠ ∈ R(λ) . Mmm (λ)

Mii (λ) = si,ν1 +...+νi−1 +1 + . . . + si,ν1 +...+νi λνi −1 + λνi and if i = j, then Mij (λ) = si,ν1 +...+νj−1 +1 + . . . + si,ν1 +...+νj λνj ; i, j = 1, . . . , m. Now let S = Ac + Bc KCc . Then sij = αij − ki cj , where ki is a row of the matrix K, cj is a column of the matrix Cc ; i = 1, . . . , m; j = 1, . . . , n. Taking into account the last remarks we transform the matrix M (λ) in the following way: ⎛⎛

α11 ⎜⎜ α21 ⎜⎜ ⎜⎜ . ⎝⎝ .. αm1

... ...

α1ν1 α2ν1 .. . αmν1

...

⎛

k1 c 1 ⎜ k2 c 1 ⎜ −⎜ . ⎝ .. km c 1 ⎛

1

⎜ =⎝

.. 0

.

... ... ...

⎞ ⎛ 1 1 0⎟ ⎜ λ ⎟ ⎜ .. ⎟ · ⎜ .. .⎠ ⎝ . 0 λν1 k1 cν1 k2 cν2 .. . km cνm

⎛ ⎞   α1,n−νm +1   ⎟ ⎜ α2,n−νm +1 ⎟ ⎜ ⎟ . . . ⎜ .. ⎝ ⎠ .     α

⎞ ⎛ 1 0 0⎟ ⎜ λ ⎟ ⎜ .. ⎟ · ⎜ .. .⎠ ⎝ . 0 λν1

⎛⎛ α11  ⎞ ⎜⎜ α21 0  k1 ⎜⎜   . ⎟ ⎜⎜ .  .. ⎠ · ⎜⎜ .. ⎜⎜  ⎝⎝ α 1  km m1 −c1

⎛  α1,n−νm +1  ⎜ α2,n−νm +1 ⎜ ⎜ .. . . . ⎜ . ⎜ ⎝ α m,n−νm +1   −c n−νm +1

... ... ... ...

m,n−νm +1

α1,n α2,n .. . αm,n −cn

... ... ... ...

... ... ...

α1,n α2,n .. . αm,n

⎛ ⎞   k1 cn−νm +1   ⎟ ⎜ k2 cn−νm +1 ⎟ ⎜ ⎟ . . . ⎜ .. ⎝ ⎠ .     k c

m n−νm +1

α1ν1 α2ν1 .. . αmν1 −cν1

⎞ 0 ⎛ 1 0⎟ ⎟ ⎜ λ .. ⎟ · ⎜ ⎜ . .⎟ ⎟ ⎝ .. ⎠ 1 λνm 0

⎞ 1 ⎛ 1 0⎟ ⎟ ⎜ λ .. ⎟ · ⎜ ⎜ . .⎟ ⎟ ⎝ .. 0⎠ λν1 0

... ... ...

⎞ ⎛ 1 0 0⎟ ⎜ λ ⎟ ⎜ .. ⎟ · ⎜ .. .⎠ ⎝ . 1 λνm k1 c n k2 c n .. . km cn

⎞ ⎟ ⎟ ⎟− ⎠

⎞ ⎛ 1 0 0⎟ ⎜ λ ⎟ ⎜ .. ⎟ · ⎜ .. .⎠ ⎝ . 0 λνm

⎞⎞ ⎟⎟ ⎟⎟ ⎟⎟ = ⎠⎠

 ⎞   ⎟ ⎟ ⎟ . . . ⎠   

⎞

⎞

⎟ ⎟⎟ ⎟⎟ ⎟⎟ = W · D(λ), ⎠⎟ ⎠

where W = (Im |K) ∈ Rm×(m+p) . Thus, we have det(λn In − Ac − Bc KCc ) = det(W · D(λ)), and for the calculation of the magnitude det(W · D(λ)) it is necessary to apply the known in Theory of Matrices the Binet–Cauchy formula. 2 Let λn +a1 λn−1 +. . .+an be the polynomial with which the characteristic polynomial det(λEn −Ac −Bc KCc ) of the closed-loop system must coincide. Then from Theorem 3.1 it follows that

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

213

⎛

⎛ ⎞ ⎞ λn λn ⎜ . ⎟ ⎜ . ⎟ ⎜ .. ⎟ ⎜ .. ⎟ m m m ⎜ ⎜ ⎟ ⎟. ∧ (Im |K) · ∧ D(λ) = ∧ (Im |K) · F · ⎜ ⎟ = (1, a1 , . . . , an ) · ⎜ ⎟ ⎝ λ ⎠ ⎝ λ ⎠ 1 1 Now taking into account a specific character of the matrix F , we have ⎛

⎞ λn ⎜ . ⎟ ⎜ .. ⎟ m m ⎟ ∧ (Im |K) · ∧ D(λ) − (1, a1 , . . . , an ) · ⎜ ⎜ ⎟ ⎝ λ ⎠ 1  ⎛ ⎞ ⎛ ⎞ 1 a ... a λn 1 n   ⎜ ⎟ ⎜ ⎟  0 0 . . . 0 ⎟ ⎜ ... ⎟ ⎜ m  ⎜ ⎜ ⎟ ⎟ = 0. = ∧ (Im |K) · ⎜ F −  . . .. ⎟ · ⎜ ⎟  .. .. . ⎠ ⎝ λ ⎠ ⎝  0 0 ... 0 1

(13)

From here it follows that for an explicit exposition of the system of equations is determining the feedback design problem solution, it is necessary to give n various real values μ1 , . . . , μn by the variable λ and to substitute them in (13). Then we derive ∧m (Im |K) · H = (1, a1 , . . . , an ) · U,

(14)

where H = (∧m D(μ1 ), . . . , ∧m D(μn )) is a real (Bin(m + p, m) × n) matrix, and ⎛

μn1 ⎜ . ⎜ .. U =⎜ ⎜ ⎝ μ1 1

⎞ . . . μnn .. ⎟ . ⎟ ⎟ ∈ R(n+1)×n ⎟ . . . μn ⎠ ... 1

is a matrix of rank n. Rewrite system (14) in the following form: ∧m (Im |K) · H1 ≡ 0, where

(15)

214

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

⎛

⎛

(1, a1 , . . . , an ) · U ⎜ ⎜ ⎜ 0......0 ⎜ ⎜ H1 = ⎜ .. .. ⎜H −⎜ . . ⎝ ⎝ 0......0

⎞⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎠⎠

(16)

is a real (Bin(m + p, m) × n) matrix and the zero matrix located under the row (1, a1 , . . . , an ) · U has sizes (Bin(m + p, m) − 1) × n. Let L1 ∈ Rm×m and L2 ∈ Rm×p be two matrices such that the matrix (L1 |L2 ) ∈ Rm×(m+p) is a solution of the following equation ∧m (L1 |L2 ) · H1 = 0.

(17)

Assume that det L1 = 0. Then from here it follows that K = L−1 1 L2 . Definition 3.1. If the solution (L1 |L2 ) of system (17) satisfies the condition det L1 = 0, then this solution is called regular; if we have det L1 = 0, then such solution is called degenerative. The matrix K resolving the feedback design problem is also called a regular solution of this problem. 4. Search method of degenerative solutions of system (17) Further, we will suppose that mp > n, m < n, p < n. Consider two cases: even one of the numbers n/m or n/p is not integer (in this situation it is enough to solve the feedback design problem for the case n = mp − 1); the numbers n/m and n/p are integer. 4.1. Case n = mp − 1 Let k = [n/m] = [(mp − 1)/m] = p − 1 be the integer part of the number n/m. (We suppose that the number n/m is not integer.) From here it follows that n − km = m − 1 and ν1 = . . . = νm−1 = p, νm = p − 1. Consider the matrix ⎛

Wm

α1,n−νm +1 α2,n−νm +1 .. .

⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ αm−1,n−νm +1 ⎜ ⎝ αm,n−νm +1 −cn−νm +1

... ...

α1,n α2,n .. .

... . . . αm−1,n ... αm,n ... −cn

⎞ 0 0⎟ ⎟ .. ⎟ .⎟ ⎟ ∈ R(m+p)×(p−1) . ⎟ 0⎟ ⎟ 1⎠ 0

(18)

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

Introduce the following inverse matrix

 T11 T = ∈ R(m+p)×(m+p) . T21 |0m |T22

215

(19)

Here T11 ∈ Rp×(m+p) , T21 ∈ Rm×(m−1) , T22 ∈ Rm×p , and the zero column 0m has dimension m. Choose matrices T11 , T21 , and T22 so that the conditions det T = 0 and ⎞ ⎛ α1,n−νm +1 ... α1,n ⎟ ⎜ ... α2,n ⎟ ⎜ α2,n−νm +1 ⎟ − T22 · (cn−νm +1 . . . cn ) = 0 ∈ Rm×(p−1) (20) T21 · ⎜ .. .. ⎟ ⎜ . ... . ⎠ ⎝ αm−1,n−νm +1 . . . αm−1,n would be satisfied. Since we have det T = 0, then rows of the matrix (T21 |0m |T22 ) ∈ Rm×(m+p) must be linearly independent. There are m + p rows in the matrix Wm and each of these rows (with the exception of m-th row) has dimension equal to k = p − 1. Thus, for solvability of the linear matrix equation (20) the inequality m + p − 1 − k ≤ m must be valid. At k = p − 1 the last inequality is justified. Therefore, a generic case equation (20) is always solvable. It is easy to check that

 Wpm T Wm = ∈ R(m+p)×m , (21) 0m×m where Wpm ∈ Rp×m and 0m×m is the zero (m × m) matrix. We will assume that matrix (18) with the help of matrix (19) is reduced to form (21). Consider the vector ∧m (T D(λ)), where D(λ) is matrix (12). Then according to condition (21) the last coordinate of this vector is equal to zero. From here it follows that ∧m Q · ∧m (T D(λ)) ≡ 0,

(22)

where Q = (0m×p |Im ) ∈ Rm×(m+p) and 0m×p is the zero (m × p) matrix. (In the matrix Q the matrix Im it is possible to replace by any real invertible matrix of degree m.) Since equality (22) is valid for any real λ, then from (12) it follows that the vector ∧m (QT ) is the solution of the system ∧m (QT ) · H ≡ 0, where the matrix H is the same as in system (14).

216

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

In system (14) we replace the matrix (Im |K) by the matrix ∧m (QT ). It is easy to check that the first coordinate of the vectors ∧m (Q) and ∧m (QT ) is equal to zero. Therefore, from here and system (14) it follows that the vector ∧m (QT ) is also a solution of the system ∧m (QT ) · H1 ≡ 0, where H1 is matrix (16). Thus, if we put (L1 |L2 ) = QT , then by definition of the matrix QT , we derive that det L1 = 0. This means that the solution QT is degenerative. 4.2. Case of integers n/m and n/p From here it follows that ν1 = . . . = νm = n/m. Therefore, we have m > n/p. Thus, we can consider that m = n/p + 1 and n = (m − 1)p (for example m = 3, p = 6, and n = 12). For solvability of equation (20) it is necessary that the condition (m −1) +p −[n/m] ≤ m (or p ≤ [n/m]) would be realized. Since mp > n, then the condition p ≤ [n/m] is always valid. Therefore, it is easy to check that in this case the matrices T and T Wm save the structure of matrices (19) and (21). Further, the case 4.2 repeats the case 4.1. 5. Iterative procedure for search of regular solutions of system (14) In this section we use the degenerative solution QT (which can be derived exactly) for finding of a regular solution for the feedback design problem by Newton’s iterative method. Introduce the matrix QL = (L | Im ) ∈ Rm×(m+p) and consider the system of equations ∧m (QL · (T + T1 )) · H1 = 0,

(23)

where L ∈ Rm×p is an unknown matrix;  is a real parameter; QL = Q, if L = 0. The matrix T1 ∈ R(m+p)×(m+p) is arbitrarily chosen satisfying a unique condition: det(T + T1 ) = 0. Denote by z1 , . . . , zn any n elements of the matrix L. The other mp − n elements of the matrix L, we will consider equal to zero. Let z = (z1 , . . . , zn )T be a column vector. Introduce the designation QL · (T + T1 ) = (G1 (, z)|G2 (, z)), where G1 (, z) ∈ Rm×m , G2 (, z) ∈ Rm×p .

(24)

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

217

Suppose that det G1 (, z) ≡ 0. Now, we rewrite system (23) in the following manner P (, z) = P (, 0) + P  (, 0) · z + {terms of degrees mi ; 2 ≤ mi ≤ m} = 0.

(25)

Here, P  (, 0) is the Jacobi matrix of the mapping P (, z) : Rn → Rn calculated in the point z = 0. Further, if  = 0, then system (23) has the solution L = 0 (system (25) has the solution z = 0). Suppose that rank P  (0, 0) = n. Then from this condition it follows that the solution z() continuously depends on the parameter  in enough small neighborhood of change of this parameter. Therefore, it is always possible to specify the value 0 ≈ 0 (|0 | = 0) of parameter  such that the solution of equation (25) z(0 ) = 0 and the first coordinate of the vector ∧m (QL · (T + 0 T1 )) will not be equal to zero. In order to obtain the solution of system (25) we use the vector z = 0 as the initial approximation for organization of the iterative process on the Newton method [7]. We will consider that det P  (0, 0) = 0. Then we have z (i+1) () = z (i) () − (P  (0, 0))−1 · P (, z (i) ()), z (0) () = 0; i = 0, 1, 2, . . .

(26)

Denote by ρ = P  (0, 0) and σ() = (P  (0, 0))−1 · P (, 0) the Euclidean norms of the matrix P  (0, 0) and the vector (P  (0, 0))−1 · P (, 0). For proof of convergence of the iterative process (26) we take advantage of the following known result adapted to our situation. Theorem 5.1. (See [7].) Let B(0, r) ∈ Rn be a ball of radius with the center at the origin of coordinates. Assume that ∀u, v ∈ B(0, r) the Jacobi matrix P  (, z) satisfies the Lipschitz condition

P  (, u) − P  (, v) ≤ Lip() · u − v , with the constant Lip() > 0. Suppose that there exists the constant  = ∗ such that the constant h(∗ ) = Lip(∗ ) · σ(∗ ) · ρ < 0.25. Denote by t1 (∗ ) and t2 (∗ ) (t1 (∗ ) < t2 (∗ )) the roots of the quadratic equation h(∗ ) · t2 − t + 1 = 0. Then in the ball B(0, r1 ) ⊂ B(0, r), where r1 ≤ t1 (∗ ) · σ(∗ ) ≤ r, equation (25) has a unique solution z ∗ and sequence (26) tends to z ∗. Proof of Theorem 5.1. Consists in the search of constant ∗ satisfying all conditions of this theorem. We have

P  (, u) − P  (, v) = (P  (0, 0) + Ω(, u, v))(u − v)

≤ ( P  (0, 0) + |Φ(, r)|) · u − v , where Ω(, u, v) : R × Rn × Rn → Rn ; Φ(, r) is a polynomial of degree m − 1. Thus, we get Lip() = ρ + |Φ(, r)|.

218

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

Further, σ() = (P  (0, 0))−1 · P (, 0) = (P  (0, 0))−1 · Σ() = |||Δ()|, where Σ() : R → Rn , Δ() is a polynomial and Δ(0) = 0. Consider the equation h · t2 − t + 1 = 0. The roots of this equation are t1 =

2 2 √ √ , t2 = . 1 + 1 − 4h 1 − 1 − 4h

Thus, if 0 < h < 0.25, then 1 < t1 < 2 < t2 < ∞. In this situation the constant h() = || · ρ · (ρ + |Φ(, r)|) · |Δ()|. Consequently, in order to fulfill all conditions of Theorem 5.1 it is necessary to find the common solution ∗ of two inequalities: || · ρ · (ρ + |Φ(, r)|) · |Δ()| < 0.25 and Δ() ≤ r. From here, we have || · ρ · (ρ + |Φ(, r)|) · r < 0.25.

(27)

Since ∀r ∈ (0, ∞) lim || · ρ · (ρ + |Φ(, r)|) · r = 0, then there exists a magnitude ∗ such →0

that at  = ∗ inequality (27) will be valid. This completes the proof of Theorem 5.1.

2

Let  = 0 and z(0 ) be a nonzero solution of system (25). Introduce the designations QL · (T + 0 T1 ) = (F1 |F2 ),

(28)

where F1 ∈ Rm×m , F2 ∈ Rm×p . Then we have det F1 = 0 (it is the first coordinate of the vector Λm (QL · (T + 0 T1 )) and the matrix K satisfying equation (14) is given by the formula K = F1−1 · F2 .

(29)

If in matrices (10) we have G = Im , then it is necessary to replace matrix (29) by the matrix G · K = G · F1−1 · F2 . Note that at  = 0 system (23) has the solution QT = (F1∗ |F2∗ ), F1∗ ∈ Rm×m , F2∗ ∈ Rm×p such that det F1∗ = 0 (it is a degenerate solution). It is natural that the solution QT can not be by solution of the feedback design problem. Therefore, the design procedure offered above was developed. It remains only to answer the following question: for what matrices A, B, and C do we have rank P  (0, 0) = n? The following theorem was proved by X. Wang. Theorem 5.2. (See [27].) If mp > n, then for the real generic matrices A, B, and C the feedback design problem always has a real solution. It means that if we define the map φ : K → (a1 , . . . , an ) depending on matrices A, B, and C by formula (14), then this map must be surjective: φ(Rm×p ) = Rn . (Precisely these matrices A, B, and C are called generic.)

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

219

It is known that in order that the map φ(K) there was surjective its necessary that the rank of Jacobi matrix ∂φ(K)/∂K at K = 0 was equal to n for all points (X, Y, Z) ∈ Σ from some open set Σ ⊂ Rn×n × Rn×m × Rp×n . We return to the matrix D(λ) (formula (12)) and its exterior degree ∧m D(λ) ∈ Rr , where r = Bin(m + p, m). Denote by Di1 ,...,im (λ) the coordinate of the vector ∧m D(λ); 1 ≤ i1 < . . . < im ≤ m + p. Select in the vector ∧m D(λ) (formula (13)) the subvector ∧m DK (λ) composed from the coordinates Di1 ,...,im (λ) such that 1 ≤ i1 < . . . < im−1 ≤ m and m + 1 ≤ im ≤ m + p. (From here it follows that dim ∧m DK (λ) = mp.) It is clear that the subvector ∧m DK (λ) generates the submatrix FK of the matrix F (see formula (13)), which is located in those rows i1 , . . . , im that the minors Di1 ,...,im (λ); here 1 ≤ i1 < . . . < im−1 ≤ m and m + 1 ≤ im ≤ m + p. Thus, we have ⎛

⎞ λn ⎜ . ⎟ ⎜ .. ⎟ m ⎟ , FK ∈ Rmp×(n+1) , ∧ DK (λ) = FK · ⎜ ⎜ ⎟ ⎝ λ ⎠ 1 and (∂φ(K)/∂K)|K=0 = FK . (Note that the first column of the matrix FK is zero.) From here it follows that if rank FK = n,

(30)

then Theorem 5.2 will be satisfied. The following theorem is a main result in this section. Theorem 5.3. Let mp > n, m < n, p < n. Then the conditions (4), (30) are sufficient for solvability of the feedback design problem. Proof. Theorem 5.3 is a corollary of Theorem 5.2. Indeed, equality (30) is an existence condition of some open set in Rmp×n . In this case, there is a submatrix of rank n of matrix FK , which can be a candidate on the role of the Jacobi matrix of system (17). It means the existence in system (25) required matrix P  (, 0). 2 6. Linear output feedback constructive design algorithm for case mp > n Now we are ready to represent the design algorithm of linear output feedback (3) for system (1) in the case rank B = m, rank C = p, and mp > n. Let λn + a1 λn−1 + . . . + an ; a1 , . . . , an ∈ R be the desired characteristic polynomial, which must have system (1) closed by the feedback (3).

220

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

The design algorithm consists of the following three parts. 6.1. Construction of Bezout’s states column model (a1) Verify the conditions (4), (5), and (6). (a2) By formula (7) compute the numbers k and l, and put ν1 = ν2 = . . . = νl = k + 1, νl+1 = νl+2 = . . . = νm = k. (a3) By the permutation matrix G fulfill the permutation {t1 , . . . , tm } of columns b1 , . . . , bm in the matrix B so that vectors (8) would be linearly independent. (a4) Solve the system of linear algebraic equations (9). (a5) By the matrices Z and G, reduce triple of matrices (A, B, C) to forms (10) and (11). 6.2. Obtaining degenerate solution for feedback design problem (a6) With the help of matrix (12) construct the matrices H, U , and system (17). (a7) Construct the matrix FK and verify condition (30). (If this condition is not fulfilled, then it is necessary to complete the algorithm.) (a8) Construct matrices (19) and QT . 6.3. Obtaining regular solution for feedback design problem (a9) Construct system (25) and verify the condition det P  (0, 0) = 0. (If this condition is not fulfilled, then it is necessary to go back to item (a3).) (a10) With the help of the matrix QT solve system (25) by iterative method (26). (Thus, (0) (0) the vector (z1 , . . . , zn )T = (0, . . . , 0)T is an initial value in iterative procedure.) For this aim it is necessary to fix any number 0 such that the magnitude |0 | is small enough to begin the iterative process (26). If sequence (26) diverges, then diminish the magnitude ||. In this case we put  = 1 , where |1 | < |0 |, and repeat process (26). It is clear that at some small  = s such that 0 < |s | < |s−1 | < . . . < |1 | < |0 | the sequence z (0) (s ), z (1) (s ), . . . will converge to the solution z ∗ (s ) of equation (25). (a11) From equality (28) find the matrices F1 and F2 . (a12) Compute matrix of the desired output feedback K for the origin system (1), (2) by the formula K = G · F1−1 · F2 . In the following example the sequence of steps which it is necessary to do for the iterative solution of the feedback design problem with the constructive choice of the initial approximation will be shown. Example 6.1. Let n = 3, m = p = 2. Suppose that system (1), (2) is given by the following matrices:

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

⎛

0 ⎜ A = ⎝ −1 1

⎞ ⎛ 1 0 0 ⎟ ⎜ −2 1⎠, B = ⎝1 0 −1 0

⎞

0 −1 −3 ⎟ 0 ⎠, C = −2 1 1

−1 −1

221

 .

Assume that the spectrum of the matrix A + BKC must consist of the eigenvalues λ1 = −2; λ2 = λ3 = −1; so the characteristic polynomial of the closed-loop system looks like λ3 + 4λ2 + 5λ + 2. Thus, in this case items (a1)–(a5) are already fulfilled. The results of implementation of items (a6)–(a8) will be: the polynomial matrix (12) is ⎛

(1 + λ)2 ⎜ −1 ⎜ D(λ) = ⎜ ⎝ 1 + 3λ 2−λ

⎞ −1 1+λ⎟ ⎟ ⎟; 1 ⎠ 1

the matrix H and equation (14) are ⎛

0 ⎜ 2 ⎜ k12 ⎜ ⎜ 3 ⎜ k22 ⎜ −2 ⎜ ⎝ −3 −1

∧2

1 0 k11 0 1 k21

7 8 5 −9 −3 3

⎞ −1 ⎛ 0 −2 ⎟ ⎟ ⎜ ⎟ 3⎟ ⎜0 ⎟ = (1 4 5 2) ⎜ ⎝0 −1 ⎟ ⎟ 1 −1 ⎠ −5

1 1 1 1

⎞ −1 1⎟ ⎟ ⎟ −1 ⎠ 1

(here as the numbers μ1 , μ2 and μ3 for construction of the matrix U numbers 0, 1, and −1 are taken accordingly); the matrix FK is ⎛

FK

⎞ 0 1 5 2 ⎜0 1 1 3⎟ ⎜ ⎟ =⎜ ⎟ , rank FK = 3 ⎝ 0 −3 −4 −2 ⎠ 0 1 −1 −3

(the conditions of Theorem 5.3 are fulfilled); the matrices (19) and QT are ⎛

1 ⎜0 ⎜ T =⎜ ⎝1 1

0 1 0 0

0 0 1 0

⎞ 0

 0⎟ 1 0 1 0 ⎟ . ⎟ , QT = 0⎠ 1 0 0 1 1

The results of implementation of items (a9)–(a12) will be:

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

222

the matrix T1 and the matrix equation (23) are ⎛

0 ⎜ 0 ⎜ T1 = ⎜ ⎝  2

0 0 0 0 0 0 0 0

⎛

⎞

0

0⎟ l11 +  + 1 ⎟ ⎟ , ∧2 l21 + 2 + 1 0⎠ 0

l12 l22

−2 −5 ⎜ 2 8 ⎜ 5 1 0 ⎜ ⎜ 3 ⎜ 0 1 ⎜ −2 −9 ⎜ ⎝ −3 −3 −1 3

⎞ −1 −2 ⎟ ⎟ 3⎟ ⎟ ⎟ = 0; −1 ⎟ ⎟ −1 ⎠ −5

let z1 = l11 , z2 = l12 , z3 = l22 , and l21 = 0; then system (25) is ⎛

3 ⎜ ⎝5 3

⎞⎛ ⎞ ⎛ ⎞ ⎛ −1 0 z1 2 0 ⎟⎜ ⎟ ⎜ ⎟ ⎜ 2 4 ⎠ ⎝ z2 ⎠ − ⎝ 5 ⎠ z1 z3 +  ⎝ 0 1 0 z3 0 0

⎞ ⎞⎛ ⎞ ⎛ z1 1 4 −2 ⎟ ⎟⎜ ⎟ ⎜ 10 −5 ⎠ ⎝ z2 ⎠ −  ⎝ 11 ⎠ = 0 −7 2 −1 z3

(here the matrix.) ⎛

3 ⎜ P (0, 0) = ⎝ 5 3

⎞ −1 0 ⎟ 2 4 ⎠ , det P (0, 0) = 0); 0 0

system (26) is ⎛

⎞ ⎛ (i+1) z1 0 −2 ⎜ ⎜ (i+1) ⎟ 6 ⎝ z2 ⎠ = ⎝0 3 (i+1) 0 −8 z3

⎞ ⎛ (i) ⎞ ⎞ ⎞ ⎛ ⎛ z1 1 1 7 ⎟⎜ ⎟ (i) (i) 1 ⎜ ⎟ ⎟ 1⎜ −3 ⎠ ⎝ z2(i) ⎠ + ⎝ −3 ⎠ z1 z2 −  ⎝ 24 ⎠ . 3 3 (i) 4 4 −29 z3

An accuracy of computations δ = 0.0001 was given and the computations at different values  equal accordingly 0.1; 0.01; 0.001 and 0.0001 were produced. The vector (0) (0) (0) (z1 , z2 , z3 )T = (0, 0, 0)T was the vector of initial values. By formula (28) the matrices F1 , F2 were obtained. After that by formula (29) the feedback matrix K was also computed (here G = I2 ). As a result the following matrices of output feedbacks were derived:

 0.5346 0.4398 K1 ( = 0.1) = , −0.6216 0.4574

 0.5462 0.4519 K2 ( = 0.01) = , −5.7381 5.5514

 0.5470 0.4527 K3 ( = 0.001) = , −56.6726 56.4951

 0.5471 0.4528 K4 ( = 0.0001) = . −566.1178 565.9291

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

223

Note that the number of iterations done for achievement of the required accuracy is accordingly such: 9, 3, 2, and 2. 7. Examples Example 7.1. Let n = 5, m = 2, and p = 3. Suppose that system (1), (2) is given by the following matrices: ⎞ 1 −2 4 2 1 ⎜2 3 −1 −3 0 ⎟ ⎟ ⎜ ⎟ ⎜ A = ⎜1 1 2 2 3⎟, B ⎟ ⎜ ⎝0 1 −2 −3 1 ⎠ 2 −2 3 −1 1 ⎛ 3 4 −2 −3 ⎜ C = ⎝ 2 1 −2 −1 1 1 3 0 ⎛

⎛

1 ⎜ −2 ⎜ ⎜ =⎜ 2 ⎜ ⎝ 1 −1 ⎞ 1 ⎟ 2⎠. 0

⎞ 2 3⎟ ⎟ ⎟ 0⎟, ⎟ 1⎠ 1

The desired roots of the characteristic polynomial of the closed-loop system are such: λ1 = −1|, λ2 = −2, λ3 = −3, λ4 = −4, λ5 = −5 (from here the characteristic polynomial follows: λ5 +15λ4 +85λ3 +225λ2 +274λ1 +120). Depending on magnitude  the following matrices of output feedbacks were derived:

 93.8541 −110.4413 −102.3904 K1 (0.001) = ; 820.6664 −964.6943 −894.6283

 13.4996 −15.9896 −14.8941 K2 (0.005) = ; 108.2339 −127.2876 −118.8808

 8.8817 −10.5601 −9.8655 K3 (0.008) = ; 67.2681 −79.1318 −74.2734

 7.3434 −8.7509 −8.1902 K4 (0.01) = . 53.6125 −63.0778 −59.4035 Example 7.2. Let n = 7, m = 2, and p = 4. Suppose that system (1), (2) is given by the following matrices: ⎞ ⎛ ⎞ 0 1 1 2 0 0 0 −1 −1 ⎜0 ⎜ 0 0⎟ 0 0 0 1 0 0⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 −1 0 0 0 1 1⎟ ⎜ 1 −1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ A = ⎜ −1 −1 2 1⎟, 0 1 0 0⎟, B = ⎜0 ⎟ ⎜ ⎟ ⎜ ⎜1 ⎜ 1 1⎟ 1 1 2 0 0 0⎟ ⎟ ⎜ ⎟ ⎜ ⎝0 ⎝ −1 0⎠ 0 0 −1 0 −1 0⎠ 1 0 0 0 1 1 0 0 1 ⎛

224

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

⎛

⎞ 1 2 −2 0 0 0 1 ⎜ 0 0 0 1 1 1 2⎟ ⎜ ⎟ C=⎜ ⎟. ⎝ −1 −1 2 2 0 0 0⎠ 1 0 1 0 −1 −1 1 The desired roots of the characteristic polynomial of the closed-loop system are such: λ1 = . . . = λ7 = −1 (from here the characteristic polynomial follows: λ7 + 7λ6 + 21λ5 + 35λ4 + 35λ3 + 21λ2 + 7λ1 + 1). Depending on magnitude  the following matrices of output feedbacks were derived:

 −70.4900 25.3175 −71.3177 117.9691 K1 (0.1) = ; −44.6268 16.2446 −45.3924 73.7786

 −54.5552 18.5602 −55.1915 88.8800 K2 (0.15) = ; −33.4474 11.5880 −34.1114 53.6777

 −46.7602 15.1765 −47.3165 74.5316 K3 (0.2) = ; −27.8327 9.2344 −28.4651 43.6240

 −42.2041 13.1420 −42.7244 66.0613 K4 (0.25) = . −24.4461 7.8055 −25.0724 37.5883 In addition, the maximum number of iterations (near 25) were required at  = 0.25. Note that in this example the desired accuracy of computations was not reached. (Nevertheless, ∀i ∈ {1, . . . , 4} the matrix A + BKi C of the closed-loop system is stable.) This fact can be explained by three reasons: at reduction of the triple matrices (A, B, C) to form (10), (11) calculating errors can be large enough; the matrix L1 in equation (17) is an improperly stipulated; ∀i ∈ {1, . . . , 4} the matrix A + BKi C is similar to the Jordan block of degree 7 (the Jordan block of arbitrary degree is not diagonalizable). 8. Conclusion The analysis of the above described examples allows us to do the following conclusions: (c1) at fixed  the feedback design problem solution does not depend on real numbers μi in the matrix U in formula (14); i = 1, . . . , n; (c2) at  || → 0 a number of iterations diminishes, however the Euclidean norm K = tr(K T K) of the regular solution K increases and it passes to the degenerate solution, which is unacceptable for the feedback design problem; (c3) if || increases then the norm of the regular solution diminishes and this solution departs from the degenerate solution. However, in this case the number of iterations increases and it can result to divergence of the iterative process (26); (c4) if mp − n > 1, then additional possibilities for an optimization of the design algorithm appear;

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

225

(c5) if to do replacements A → AT , B → C T , C → B T , then all results described above remain valid. (In this case condition (4) in Theorem 5.3 must be substituted by condition (5).) Besides, the assignment of a number lij equal to zero in the matrix L of equation (23) and suppositions about the form of the matrix T1 give us some freedom of choice. These remarks abandon a wide field of activity for an improvement of the offered design algorithm. Another important problem that is not considered in the present work is a robustness problem of solutions, which are derived by the algorithm of Section 6. The robustness problem is a complex problem. Therefore, we will consider only one aspect. Let ΔA ∈ Rn×n , ΔB ∈ Rn×m , and ΔC ∈ Rp×n be some perturbations of matrices A, B, and C. Assume that the feedback controller K ∈ Rm×p provides stability of the closed-loop system ˙ x(t) = (A + BKC)x(t).

(31)

We use the same controller for providing stability of the perturbed closed-loop system ˙ x(t) = (A + ΔA + (B + ΔB)K(C + ΔC))x(t).

(32)

In order to provide stability of system (32) it is necessary that the norm of the matrix P (ΔA, ΔB, ΔC) = (A + ΔA + (B + ΔB)K(C + ΔC)) − (A + BKC) = ΔA + ΔBKC + BKΔC + ΔBKΔC would be small enough. Let Q = tr(QT Q) be the Euclidean norm of a square matrix Q. Introduce the balls: BA = {ΔA ∈ Rn×n | ΔA ≤ SA }, BB = {ΔB ∈ Rn×m | ΔB ≤ SB }, and BC = {ΔC ∈ Rp×n | ΔC ≤ SC }. Assume that for the fixed matrices A, B, C, and K, and any perturbations ΔA ∈ BA , ΔB ∈ BB , and ΔC ∈ BC systems (31) and (32) are stable. Definition 8.1. The positive number

sm(K, A, B, C) =

max

0≤SA ,SB ,SC <∞

 max

ΔA∈BA ,ΔB∈BB ,ΔC∈BC

P (ΔA, ΔB, ΔC)

is called a stability margin of system (1), (2) with feedback controller K. Thus, under robustness of the feedback controller K, we understand the stability margin of system (31) with the feedback controller K. (Note that sm(K, A, B, C) < ∞.

226

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

Really, if a square matrix D is stable, then the perturbed matrix D + ΔD can remain continuously stable in the ball BD = {ΔD ∈ Rn×n | ΔD ≤ SD } only if SD < ∞.) Let K1 and K2 be two different regular solutions of the feedback design problem: λn + a1 (K1 )λn−1 + . . . + an (K1 ) ≡ λn + a1 (K2 )λn−1 + . . . + an (K2 ). If sm(K1 , A, B, C) < sm(K2 , A, B, C), then the robustness of feedback controller K2 is more than the robustness of feedback controller K1 . According to the algorithm of Section 6 the regular solution of the feedback design problem is a function depending on . Then the robustness problem is reduced to the maximization problem of the function sm(K(), A, B, C). Solution methods of this problem substantially differ from the methods of the present article. Therefore, in the present paper the robustness problem was not solved. It was said in Section 1 that there exists the LMI method (see [14]), which in most important in control theory cases allows to solve the output feedback design problem. However, there is a situation (see [14], subsection 3.2), at which the design problem can not be solved by the LMI method. In this case it is possible to apply the algorithm of Section 6 of the present paper. Then the derived output feedback matrix K can be used for the second step of the two-step strategy, which was described in the paper [14]. Acknowledgements The author is grateful to the anonymous referees for the brilliant comments which led to a significant improvement of the presentation. References [1] V.Ye. Belozyorov, On the solutions of a modal control problem in a limiting case, J. Autom. Inf. Sci. 32 (5) (2000) 20–28. [2] V.Ye. Belozyorov, On invariant design of feedback for bilinear control systems of second order, Int. J. Appl. Math. Comput. Sci. 11 (2) (2001) 101–113. [3] V.Ye. Belozyorov, Design of linear feedback for bilinear control systems, Int. J. Appl. Math. Comput. Sci. 12 (4) (2002) 493–511. [4] I. Blumthaler, U. Oberst, Design, parametrization, and pole placement of stabilizing output feedback compensators via injective cogenerator quotient signal modules, Linear Algebra Appl. 436 (2012) 963–1000. [5] R. Brockett, C.I. Byrnes, Multivariable Nyquist criteria, root loci, and pole placement: a geometric viewpoint, IEEE Trans. Automat. Control AC-26 (1) (1981) 271–284. [6] Y.-Y. Cao, J. Lam, Y.-X. Sun, Static output stabilization: an ILMI approach, Automatica 34 (12) (1998) 1641–1645. [7] J.E. Dennis, R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Inc., Englewood Cliffs, NJ, USA, 1983. [8] L.R. Fletcher, J.F. Magni, Exact pole assignment by output feedback, Internat. J. Control 37 (6) (1987) 1955–2033. [9] V. Jurdjevic, J. Quinn, Controllability and stability, J. Differential Equations 28 (1978) 381–389. [10] H. Kimura, Pole placement by gain output feedback, IEEE Trans. Automat. Control AC-20 (4) (1975) 509–516. [11] U. Konigorski, Pole placement by parametric output feedback, Systems Control Lett. 61 (2) (2012) 292–297. [12] D.G. Luenberger, Canonical forms for linear multivariable systems, IEEE Trans. Automat. Control AC-12 (1) (1967) 290–293.

V.Ye. Belozyorov / Linear Algebra and its Applications 504 (2016) 204–227

227

[13] J.F. Magni, C. Champetier, A geometric framework for pole assignment algorithms, IEEE Trans. Automat. Control AC-36 (9) (1991) 1105–1111. [14] F. Palacios-Quinonero, J. Rubio-Massegu, J.M. Rossell, H.R. Karimi, Recent advances in static output-feedback controller design with applications to vibration control of large structures, Model. Identif. Control 35 (3) (2014) 169–190. [15] L. Praly, Z.P. Jiang, Linear output feedback with dynamic high gain for nonlinear systems, Systems Control Lett. 53 (10) (2004) 107–116. [16] M.S. Ravi, J. Rosenthal, X. Wang, Dynamic pole assignment and Schubert calculus, SIAM J. Control Optim. 34 (3) (1996) 813–832. [17] M.S. Ravi, J. Rosenthal, U. Helmke, Output feedback invariants, Linear Algebra Appl. 351 (352) (2002) 623–637. [18] J. Rosenthal, X. Wang, Output feedback pole placement with dynamic compensator, IEEE Trans. Automat. Control AC-41 (2) (1996) 830–843. [19] D. Rosinova, V. Vesely, V. Kucera, A necessary and sufficient condition for static output feedback stabilizability of linear discrete-time systems, Kybernetika 39 (2003) 447–459. [20] V.L. Syrmos, C.T. Abdallah, P. Dorato, K. Grigoriadis, Static output feedback – a survey, Automatica 33 (2) (1997) 125–137. [21] S. Skogestad, I. Postlethwaite, Multivariable Feedback Control – Analysis and Design, Wiley, New York, 2005. [22] E.D. Sontag, Mathematical Control Theory, Springer-Verlag, New York, 1990. [23] C. Scherer, P. Gahinet, M. Chilali, Multiobjective output-feedback control via LMI optimization, IEEE Trans. Automat. Control AC-42 (7) (1997) 896–911. [24] A. Vanecek, Models: equivalence, uses, extensions, in: P. Eykhoff (Ed.), Trends and Progress in System Identification, Pergamon Press, Oxford, 1981, pp. 29–66. [25] V. Vesely, Static output controller design, Kybernetika 37 (2) (2001) 205–221. [26] X. Wang, On output feedback via Grassmannians, SIAM J. Control Optim. 19 (4) (1991) 926–935. [27] X. Wang, Pole placement by static output feedback, J. Math. Syst. Estim. Control 2 (2) (1992) 205–218. [28] J.C. Willems, Input-output and state-space representations of finite-dimensional linear timeinvariant systems, Linear Algebra Appl. 50 (1993) 581–608. [29] W.M. Wonham, On pole assignment in multi-input controllable systems, IEEE Trans. Automat. Control AC-12 (6) (1967) 660–665. [30] W.M. Wonham, Linear Multivariable Control. A Geometric Approach, Springer-Verlag, New-York, 1985.