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Common Lyapunov Function in the Control Problem for Linear Dynamical Systems*
15th IFAC Workshop on Control Applications of Optimization The International Federation of Automatic Control September 13-16, 2012. Rimini, Italy
Common Lyapunov Function in the Control Problem for Linear Dynamical Systems Igor M. Ananyevskiy ∗ Alexander I. Ovseevich ∗∗ ∗
Institute for Problems of Mechanics, Russian Academy of Sciences, Moscow, Russia (e-mail: [email protected]) ∗∗ Institute for Problems of Mechanics, Russian Academy of Sciences, Moscow, Russia (e-mail: [email protected]) Abstract: A problem of designing a bounded feedback control for a linear dynamical system is considered. A feedback control law is proposed which is bounded and steers any state in a neighborhood of the origin to the origin in a finite time. The construction is based on the notion of a common Lyapunov function. The existence of a common quadratic Lyapunov function for matrices of specific forms is proven. Some remarkable properties of this function are established. Keywords: Linear dynamic systems, feedback control, finite time of motion, common Lyapunov function. 1. INTRODUCTION
a great amount of computations while our control algorithm proposed below does not require much memory or computational power. Our control is more smooth than the minimum–time one: its only singular point is zero, while the singular locus of optimal control is a singular hypersurface.
A symmetric positive definite matrix Q defines a common quadratic Lyapunov function for two stable matrices A1 and A2 if it satisfies the inequalities i = 1, 2. (1) Ai Q + QA i < 0, For matrices Ai given numerically, linear matrix inequalities (1) can be solved very efficiently by newly developed computational methods. However, for an arbitrary pair of matrices Ai there are no direct general effective conditions for the existence of a solution to inequalities (1). Below, we prove the existence of a common quadratic Lyapunov function for matrices of specific forms that appear in a certain control problem.
2. CONTROL PROBLEM Consider a linear completely controllable system ¯ ¯x + B ¯u x ¯˙ = A¯ ¯, x ¯ ∈ RN , u¯ ∈ R.
(2) We are looking for a feedback control which is bounded, and steers any state in a neighborhood of the origin to the origin in a finite time.
The starting point of the research proposed is the problem of bringing a dynamic system from one given state to another one in a finite time. In the case of a linear dynamic system, the problem can be rather easily solved by means of an open-loop control (Kalman et al. (1969)). However, we are searching for a closed-loop control because it has obvious advantages: the closed-loop control can cope with unknown disturbances and it is effective under uncertainties of parameters of the model.
The idea of our approach to the local synthesis goes back to Korobov (1979). It uses a preliminary reduction of system (2) to a canonical form by means of transformations ¯ A¯ → A¯ + BC, u¯ → u ¯ − Cx ¯ (3) corresponding to adding a linear feedback, and a gauge transformations ¯ ¯ → D−1 B. ¯ A¯ → D−1 AD, B (4) Lemma 1. (Brunovsky (1970)). By means of transformations (3) and (4) system (2) can be made into x˙ = Ax + Bu, (5) where A = ⊕Ai , B = ⊕Bi , ⎛ ⎞ 0 0 0 ... 0 ⎛ ⎞ ⎜−1 0 0 . . . 0⎟ 1 ⎜ ⎟ 0 . . . 0⎟ ⎜ 0 −2 ⎜0⎟ ⎟ , Bi = ⎜ Ai = ⎜ (6) .. . . .. ⎟ ⎜ .. .. ⎝ ... ⎠ , . .⎟ . ⎜ . . ⎟ ⎝0 0 0 . . . 0⎠ 0 0 0 −Ni + 1 . . . 0 ¯. and Ni = N
In many cases, for nonlinear dynamical systems it suffices to solve the problem locally, in a vicinity of the target point because it is often easy to reach the vicinity in finite time. If our target is an equilibrium point it is natural to linearize the system. It might happen that the feedback control for linearized system solves the initial nonlinear problem. In principle, one can get the terminal state by using the minimum time control Pontryagin et al. (1962); Kalman et al. (1969). However, time optimal control law requires Supported by the program “Leading Scientific Schools” (project No SS-369.2012.1) and by the Russian Foundation for Basic Research (grants No 11-08-00435, 11-01-00378, and 11-01-00472).
978-3-902823-14-4/12/$20.00 © 2012 IFAC
Thus, it suffices to study the system (5) with canonical matrices (6), which is a set of independent subsystems. 68
10.3182/20120913-4-IT-4027.00037
CAO 2012 September 13-16, 2012. Rimini, Italy
4. JUSTIFICATION OF THE CONTROL
The problem now is to bring each subsystem to zero by a bounded feedback control. ¿From now on we drop the index i.
4.1 Proof of Theorem 2. Statement A means that the following matrix inequalities hold 1 (13) {M, q} > 0, {A, q} − {B, B ∗ } < 0, 2 where we make use of the “Jordan brackets” {α, β} = αβ+ β ∗ α∗ . Indeed, if Q(x, x) = (Qx, x) is a quadratic Lyapunov function for a stable matrix A, then this is equivalent to the matrix inequality {Q, A∗ } < 0, or, what is the same,
3. MAIN RESULTS Following Ananievskii et al. (2010), introduce a matrix function of a positive parameter T intimately related to system (5), (6) δ(T ) = diag{T −1 , T −2 , . . . , T −N }. (7) In what follows the parameter T will be a function T = T (x) of the state vector. Introduce the matrix q 1 qij = xi+j−2 (1 − x)dx = [(i + j)(i + j − 1)]−1 , 0
(14) {A, Q−1 } = Q−1 {Q, A∗ }Q−1 < 0. 1 −1 Moreover, the matrix 2 {A, Q } corresponds to the negative quadratic form Q−1 (x, A∗ x). A trivial computation shows that 1 {BC, q} = − {B, B ∗ }. (15) 2
(8)
i, j = 1, . . . , N, and matrices Q=q
−1
1 , C = − B ∗ Q, M = diag{1, 2 . . . , N }. 2
We implement the phase space RN as the space of real polynomials f of degree less than N of a scalar variable x. The canonical basis ek of the space RN is formed then by monomials mk (x) = xk−1 . Note, that the matrix A∗ ∂ then corresponds to the differential operator f → − ∂x f, ∗ while the matrix M = M corresponds to the operator ∂ xf. The dual vector B ∗ = (1, 0, . . . , 0) corresponds f → ∂x to the functional f → f (0).
(9)
Now we define a feedback control by the formula u(x) = Cδ(T (x))x, (10) where the function T = T (x) is defined implicitly by the condition (Qδ(T )x, δ(T )x) = 1. (11)
Now we consider the relations (13) in this functional model. The quadratic form q(f, f ), corresponding to the matrix q, has the form 1 f 2 (x)(1 − x)dx. (16)
The basic result on steering canonical system (5),(6) to the origin is as follows: Theorem 2. A. The matrix Q defines a common quadratic Lyapunov function for the matrices −M and A + BC.
0
This is a positive form. The matrices {M, q}, {A, q}, and {B, B ∗ } correspond in the functional model to the following quadratic forms: µ(f ) = q(f, M ∗ f ) =
∂ xf (x)f (x)(1 − x)dx, =2 ∂x ∗ α(f ) = q(f,A f ) = (17)
∂ f (x) f (x)(1 − x)dx, = −2 ∂x 2 β(f ) = 2f (0) , where integration is over the interval [0, 1]. Partial integration gives ∂ 2 f (x)(1 − x)dx = α(f ) = − ∂x
B. Equation (11) defines T = T (x) uniquely. √ C. The control (10) is bounded: |u| ≤ 12 Q11 . D. The control (10) brings x to 0 in time T (x). Theorem 2 was obtained for the first time in Korobov et al. (2004) in a less precise form. We have found an essentially new, and a much shorter proof. Our method allows to define a large class of common Lyapunov functions for −M and A + BC. Another result is not directly related to the optimal control theory. Theorem 3. The matrix Q has even integer elements. A proof of Theorems 2, 3 is given below in the next sections.
= − f 2 (x)dx + f 2 (0), µ(f ) = (18) 2 = 2 f (x)(1 − x)dx − f 2 (x)[(1 − x)x] dx = 1 = 2 f 2 (x)[(1 − x) + (2x − 1)]dx = f 2 (x)dx. 2 Thus, 1 α(f ) − β(f ) = −µ(f ), (19) 2 and both parts of the latter equality coincide with the negative quadratic form − f 2 (x)dx. This proves inequalities (13) and the statement A of the Theorem. Moreover, we have shown that −{M, q} = {A, q} + {BC, q}. (20)
As an illustration for the above theorems consider the case N = 4. Then the matrix Q = q −1 has the following form: ⎛ ⎞ 20 −180 420 −280 ⎜ −180 2220 −5880 4200 ⎟ (12) ⎝ 420 −5880 16800 −12600 ⎠ −280 4200 −12600 9800 and the explicit form of equation (11) is: T 8 − 20x21 T 6 + 360x1 x2 T 5 − (2220x22 + 840x1 x3 )T 4 + +(11760x2x3 + 560x1 x4 )T 3 − (8400x2 x4 + 16800x23)T 2 + +25200x3x4 T − 9800x24 = 0. Computations make it plausible the following strengthening of Theorem 3: all elements Q are multiples of Q11 . 69
CAO 2012 September 13-16, 2012. Rimini, Italy
The latter relation is equivalent to coincidence of quadratic forms (Qy, [A + BC]y) = −(Qy, M y). (21)
Pn (x)xn (1 − x)dx = 1 n (−1)n (2n + 1)! n (−1) x (1 − x)n+1 dx = (31) = n!(n + 1)! 0 (2n + 1)! B(n + 2, n + 1), = n!(n + 1)! where Γ(α)Γ(β) (32) B(α, β) = Γ(α + β) is the B-function of Euler. Finally we have (2n + 1)! Γ(n + 2)Γ(n + 1) = Pn 2 dµ = n!(n + 1)! Γ(2n + 3) (33) (2n + 1)! (n + 1)!n! 1 = = . n!(n + 1)! (2n + 2)! 2(n + 1) It follows from Rodrigues’ formula (27) that Pn ∈ Z[x] 1 n is an integer polynomial, because the operator n! ∂ maps Z[x] intoitself. This fact can be rewritten in the form aij mj , where mj = xj−1 is an element of the Pi−1 = standard monomial basis, and A = (aij ) is an integer (triangular) matrix of coefficients of the Jacobi polynomials. The above formulas for scalar products can be rewritten in the form n 1 ∗ A qA = diag , (34) 2k k=1 or, what is the same Q = A diag{2k}A∗ . (35) The last formula obviously implies that Q is an even integer matrix.
Statement B follows from the strict monotonicity of the function T → (Qδ(T )x, δ(T )x), which, in turn, follows immediately from the first inequality (13). Statement C follows from the Cauchy – Bunyakovsky inequality. Indeed, 1 u = − (QB, y), (22) 2 where y = δ(T )x, and (Qy, y) = 1. Therefore, 1 |u| ≤ (Qy, y)1/2 (QB, B)1/2 ≤ 2 √ (23) Q11 1 . ≤ (QB, B)1/2 = 2 2 Statement D follows from the computation of the total derivative T˙ . Put δ = δ(T ), then d δAδ −1 = T −1 A, δB = T −1 B, δ = −T −1 M δ, (24) dT which immediately provide for y = δ(T )x the equation
y˙ = T −1 Ay + Bu − T˙ M y . (25)
It follows then from (10), (11) that (Qy, [A + BC]y − T˙ M y) = 0,
2
(26)
but in view of (21) this implies that T˙ = −1.
Pn dµ = cn
5. CONCLUDING REMARKS 4.2 Proof of Theorem 3. The control obtained is global for linear system in the canonical Brunovsky form: it is bounded in the whole phase space and brings any initial state of system (5) with canonical matrices (6) to zero in a finite time.
The result is obtained from a study of orthogonal polynomials (shifted Jacobi polynomials), related to the measure dµ = (1 − x)dx in the interval [0, 1]. The required polynomials Pn are given by Rodrigues’ formula 1 ∂ n (1 − x)(x − x2 )n , (27) Pn (x) = n!(1 − x) where ∂ =
d dx .
Indeed,
Pn (x)xm dµ = 0
The control is locally equivalent to optimal ones. This means that the time τ (x) required for our control to bring a given state x to 0 is not much greater than the minimal τ (x) one τmin (x): the ratio τmin (x) is bounded as x runs over a neighborhood of zero.
(28) REFERENCES
if m < n, since Pn (x)xm dµ = 1 = ∂ n (1 − x)(x − x2 )n xm dx = (29) n! n (−1) (1 − x)(x − x2 )n ∂ n xm dx = 0 = n! because of partial integration, and the fact that ∂ n xm = 0. Therefore, the polynomials Pn and Pm are orthogonal if n = m. It is easy to compute the higher coefficient cn of the polynomial Pn . It coincides with the higher coefficient of the polynomial (−1)n n 2n+1 πn (x) = ∂ x , (30) n!x
I.M. Ananievskii, N.V. Anokhin, and A.I. Ovseevich. Synthesis of bounded controls for linear dynamical systems by using common Lyapunov functions. Doklady Mathematics, Vol. 82, No. 2, 1–4, 2010. P. Brunovsky. A classification of linear controllable systems. Kibernetika, 6, 176–188, 1970. R.E. Kalman, P. Falb, and M. Arbib. Topics in mathematical system theory. McGraw-Hill, New York, 1969. V.I. Korobov. A general approach to the solution of the bounded control synthesis problem in a controllability problem. Mat. Sb. (N.S.), 109(151):4(8), 582–606, 1979. V.I. Korobov, V.A. Skorik, A.E. Choque Rivero. Controllability function as the duration of a motion I. Math. Phys, Analysis, Geometry, 11, no. 2, 208–225, 2004. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishechenko. The mathematical theory of optimal processes. New York/London. John Wiley & Sons, 1962.
n
(2n+1)! which is clearly equals to (−1) n!(n+1)! . Compute the squared norm of the polynomial Pn :
70
Common Lyapunov Function in the Control Problem for Linear Dynamical Systems Igor M. Ananyevskiy ∗ Alexander I. Ovseevich ∗∗ ∗
Institute for Problems of Mechanics, Russian Academy of Sciences, Moscow, Russia (e-mail: [email protected]) ∗∗ Institute for Problems of Mechanics, Russian Academy of Sciences, Moscow, Russia (e-mail: [email protected]) Abstract: A problem of designing a bounded feedback control for a linear dynamical system is considered. A feedback control law is proposed which is bounded and steers any state in a neighborhood of the origin to the origin in a finite time. The construction is based on the notion of a common Lyapunov function. The existence of a common quadratic Lyapunov function for matrices of specific forms is proven. Some remarkable properties of this function are established. Keywords: Linear dynamic systems, feedback control, finite time of motion, common Lyapunov function. 1. INTRODUCTION
a great amount of computations while our control algorithm proposed below does not require much memory or computational power. Our control is more smooth than the minimum–time one: its only singular point is zero, while the singular locus of optimal control is a singular hypersurface.
A symmetric positive definite matrix Q defines a common quadratic Lyapunov function for two stable matrices A1 and A2 if it satisfies the inequalities i = 1, 2. (1) Ai Q + QA i < 0, For matrices Ai given numerically, linear matrix inequalities (1) can be solved very efficiently by newly developed computational methods. However, for an arbitrary pair of matrices Ai there are no direct general effective conditions for the existence of a solution to inequalities (1). Below, we prove the existence of a common quadratic Lyapunov function for matrices of specific forms that appear in a certain control problem.
2. CONTROL PROBLEM Consider a linear completely controllable system ¯ ¯x + B ¯u x ¯˙ = A¯ ¯, x ¯ ∈ RN , u¯ ∈ R.
(2) We are looking for a feedback control which is bounded, and steers any state in a neighborhood of the origin to the origin in a finite time.
The starting point of the research proposed is the problem of bringing a dynamic system from one given state to another one in a finite time. In the case of a linear dynamic system, the problem can be rather easily solved by means of an open-loop control (Kalman et al. (1969)). However, we are searching for a closed-loop control because it has obvious advantages: the closed-loop control can cope with unknown disturbances and it is effective under uncertainties of parameters of the model.
The idea of our approach to the local synthesis goes back to Korobov (1979). It uses a preliminary reduction of system (2) to a canonical form by means of transformations ¯ A¯ → A¯ + BC, u¯ → u ¯ − Cx ¯ (3) corresponding to adding a linear feedback, and a gauge transformations ¯ ¯ → D−1 B. ¯ A¯ → D−1 AD, B (4) Lemma 1. (Brunovsky (1970)). By means of transformations (3) and (4) system (2) can be made into x˙ = Ax + Bu, (5) where A = ⊕Ai , B = ⊕Bi , ⎛ ⎞ 0 0 0 ... 0 ⎛ ⎞ ⎜−1 0 0 . . . 0⎟ 1 ⎜ ⎟ 0 . . . 0⎟ ⎜ 0 −2 ⎜0⎟ ⎟ , Bi = ⎜ Ai = ⎜ (6) .. . . .. ⎟ ⎜ .. .. ⎝ ... ⎠ , . .⎟ . ⎜ . . ⎟ ⎝0 0 0 . . . 0⎠ 0 0 0 −Ni + 1 . . . 0 ¯. and Ni = N
In many cases, for nonlinear dynamical systems it suffices to solve the problem locally, in a vicinity of the target point because it is often easy to reach the vicinity in finite time. If our target is an equilibrium point it is natural to linearize the system. It might happen that the feedback control for linearized system solves the initial nonlinear problem. In principle, one can get the terminal state by using the minimum time control Pontryagin et al. (1962); Kalman et al. (1969). However, time optimal control law requires Supported by the program “Leading Scientific Schools” (project No SS-369.2012.1) and by the Russian Foundation for Basic Research (grants No 11-08-00435, 11-01-00378, and 11-01-00472).
978-3-902823-14-4/12/$20.00 © 2012 IFAC
Thus, it suffices to study the system (5) with canonical matrices (6), which is a set of independent subsystems. 68
10.3182/20120913-4-IT-4027.00037
CAO 2012 September 13-16, 2012. Rimini, Italy
4. JUSTIFICATION OF THE CONTROL
The problem now is to bring each subsystem to zero by a bounded feedback control. ¿From now on we drop the index i.
4.1 Proof of Theorem 2. Statement A means that the following matrix inequalities hold 1 (13) {M, q} > 0, {A, q} − {B, B ∗ } < 0, 2 where we make use of the “Jordan brackets” {α, β} = αβ+ β ∗ α∗ . Indeed, if Q(x, x) = (Qx, x) is a quadratic Lyapunov function for a stable matrix A, then this is equivalent to the matrix inequality {Q, A∗ } < 0, or, what is the same,
3. MAIN RESULTS Following Ananievskii et al. (2010), introduce a matrix function of a positive parameter T intimately related to system (5), (6) δ(T ) = diag{T −1 , T −2 , . . . , T −N }. (7) In what follows the parameter T will be a function T = T (x) of the state vector. Introduce the matrix q 1 qij = xi+j−2 (1 − x)dx = [(i + j)(i + j − 1)]−1 , 0
(14) {A, Q−1 } = Q−1 {Q, A∗ }Q−1 < 0. 1 −1 Moreover, the matrix 2 {A, Q } corresponds to the negative quadratic form Q−1 (x, A∗ x). A trivial computation shows that 1 {BC, q} = − {B, B ∗ }. (15) 2
(8)
i, j = 1, . . . , N, and matrices Q=q
−1
1 , C = − B ∗ Q, M = diag{1, 2 . . . , N }. 2
We implement the phase space RN as the space of real polynomials f of degree less than N of a scalar variable x. The canonical basis ek of the space RN is formed then by monomials mk (x) = xk−1 . Note, that the matrix A∗ ∂ then corresponds to the differential operator f → − ∂x f, ∗ while the matrix M = M corresponds to the operator ∂ xf. The dual vector B ∗ = (1, 0, . . . , 0) corresponds f → ∂x to the functional f → f (0).
(9)
Now we define a feedback control by the formula u(x) = Cδ(T (x))x, (10) where the function T = T (x) is defined implicitly by the condition (Qδ(T )x, δ(T )x) = 1. (11)
Now we consider the relations (13) in this functional model. The quadratic form q(f, f ), corresponding to the matrix q, has the form 1 f 2 (x)(1 − x)dx. (16)
The basic result on steering canonical system (5),(6) to the origin is as follows: Theorem 2. A. The matrix Q defines a common quadratic Lyapunov function for the matrices −M and A + BC.
0
This is a positive form. The matrices {M, q}, {A, q}, and {B, B ∗ } correspond in the functional model to the following quadratic forms: µ(f ) = q(f, M ∗ f ) =
∂ xf (x)f (x)(1 − x)dx, =2 ∂x ∗ α(f ) = q(f,A f ) = (17)
∂ f (x) f (x)(1 − x)dx, = −2 ∂x 2 β(f ) = 2f (0) , where integration is over the interval [0, 1]. Partial integration gives ∂ 2 f (x)(1 − x)dx = α(f ) = − ∂x
B. Equation (11) defines T = T (x) uniquely. √ C. The control (10) is bounded: |u| ≤ 12 Q11 . D. The control (10) brings x to 0 in time T (x). Theorem 2 was obtained for the first time in Korobov et al. (2004) in a less precise form. We have found an essentially new, and a much shorter proof. Our method allows to define a large class of common Lyapunov functions for −M and A + BC. Another result is not directly related to the optimal control theory. Theorem 3. The matrix Q has even integer elements. A proof of Theorems 2, 3 is given below in the next sections.
= − f 2 (x)dx + f 2 (0), µ(f ) = (18) 2 = 2 f (x)(1 − x)dx − f 2 (x)[(1 − x)x] dx = 1 = 2 f 2 (x)[(1 − x) + (2x − 1)]dx = f 2 (x)dx. 2 Thus, 1 α(f ) − β(f ) = −µ(f ), (19) 2 and both parts of the latter equality coincide with the negative quadratic form − f 2 (x)dx. This proves inequalities (13) and the statement A of the Theorem. Moreover, we have shown that −{M, q} = {A, q} + {BC, q}. (20)
As an illustration for the above theorems consider the case N = 4. Then the matrix Q = q −1 has the following form: ⎛ ⎞ 20 −180 420 −280 ⎜ −180 2220 −5880 4200 ⎟ (12) ⎝ 420 −5880 16800 −12600 ⎠ −280 4200 −12600 9800 and the explicit form of equation (11) is: T 8 − 20x21 T 6 + 360x1 x2 T 5 − (2220x22 + 840x1 x3 )T 4 + +(11760x2x3 + 560x1 x4 )T 3 − (8400x2 x4 + 16800x23)T 2 + +25200x3x4 T − 9800x24 = 0. Computations make it plausible the following strengthening of Theorem 3: all elements Q are multiples of Q11 . 69
CAO 2012 September 13-16, 2012. Rimini, Italy
The latter relation is equivalent to coincidence of quadratic forms (Qy, [A + BC]y) = −(Qy, M y). (21)
Pn (x)xn (1 − x)dx = 1 n (−1)n (2n + 1)! n (−1) x (1 − x)n+1 dx = (31) = n!(n + 1)! 0 (2n + 1)! B(n + 2, n + 1), = n!(n + 1)! where Γ(α)Γ(β) (32) B(α, β) = Γ(α + β) is the B-function of Euler. Finally we have (2n + 1)! Γ(n + 2)Γ(n + 1) = Pn 2 dµ = n!(n + 1)! Γ(2n + 3) (33) (2n + 1)! (n + 1)!n! 1 = = . n!(n + 1)! (2n + 2)! 2(n + 1) It follows from Rodrigues’ formula (27) that Pn ∈ Z[x] 1 n is an integer polynomial, because the operator n! ∂ maps Z[x] intoitself. This fact can be rewritten in the form aij mj , where mj = xj−1 is an element of the Pi−1 = standard monomial basis, and A = (aij ) is an integer (triangular) matrix of coefficients of the Jacobi polynomials. The above formulas for scalar products can be rewritten in the form n 1 ∗ A qA = diag , (34) 2k k=1 or, what is the same Q = A diag{2k}A∗ . (35) The last formula obviously implies that Q is an even integer matrix.
Statement B follows from the strict monotonicity of the function T → (Qδ(T )x, δ(T )x), which, in turn, follows immediately from the first inequality (13). Statement C follows from the Cauchy – Bunyakovsky inequality. Indeed, 1 u = − (QB, y), (22) 2 where y = δ(T )x, and (Qy, y) = 1. Therefore, 1 |u| ≤ (Qy, y)1/2 (QB, B)1/2 ≤ 2 √ (23) Q11 1 . ≤ (QB, B)1/2 = 2 2 Statement D follows from the computation of the total derivative T˙ . Put δ = δ(T ), then d δAδ −1 = T −1 A, δB = T −1 B, δ = −T −1 M δ, (24) dT which immediately provide for y = δ(T )x the equation
y˙ = T −1 Ay + Bu − T˙ M y . (25)
It follows then from (10), (11) that (Qy, [A + BC]y − T˙ M y) = 0,
2
(26)
but in view of (21) this implies that T˙ = −1.
Pn dµ = cn
5. CONCLUDING REMARKS 4.2 Proof of Theorem 3. The control obtained is global for linear system in the canonical Brunovsky form: it is bounded in the whole phase space and brings any initial state of system (5) with canonical matrices (6) to zero in a finite time.
The result is obtained from a study of orthogonal polynomials (shifted Jacobi polynomials), related to the measure dµ = (1 − x)dx in the interval [0, 1]. The required polynomials Pn are given by Rodrigues’ formula 1 ∂ n (1 − x)(x − x2 )n , (27) Pn (x) = n!(1 − x) where ∂ =
d dx .
Indeed,
Pn (x)xm dµ = 0
The control is locally equivalent to optimal ones. This means that the time τ (x) required for our control to bring a given state x to 0 is not much greater than the minimal τ (x) one τmin (x): the ratio τmin (x) is bounded as x runs over a neighborhood of zero.
(28) REFERENCES
if m < n, since Pn (x)xm dµ = 1 = ∂ n (1 − x)(x − x2 )n xm dx = (29) n! n (−1) (1 − x)(x − x2 )n ∂ n xm dx = 0 = n! because of partial integration, and the fact that ∂ n xm = 0. Therefore, the polynomials Pn and Pm are orthogonal if n = m. It is easy to compute the higher coefficient cn of the polynomial Pn . It coincides with the higher coefficient of the polynomial (−1)n n 2n+1 πn (x) = ∂ x , (30) n!x
I.M. Ananievskii, N.V. Anokhin, and A.I. Ovseevich. Synthesis of bounded controls for linear dynamical systems by using common Lyapunov functions. Doklady Mathematics, Vol. 82, No. 2, 1–4, 2010. P. Brunovsky. A classification of linear controllable systems. Kibernetika, 6, 176–188, 1970. R.E. Kalman, P. Falb, and M. Arbib. Topics in mathematical system theory. McGraw-Hill, New York, 1969. V.I. Korobov. A general approach to the solution of the bounded control synthesis problem in a controllability problem. Mat. Sb. (N.S.), 109(151):4(8), 582–606, 1979. V.I. Korobov, V.A. Skorik, A.E. Choque Rivero. Controllability function as the duration of a motion I. Math. Phys, Analysis, Geometry, 11, no. 2, 208–225, 2004. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishechenko. The mathematical theory of optimal processes. New York/London. John Wiley & Sons, 1962.
n
(2n+1)! which is clearly equals to (−1) n!(n+1)! . Compute the squared norm of the polynomial Pn :
70