Lecture 15: H Control Synthesis

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Transcription:

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 1/14 Lecture 15: H Control Synthesis Example

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 1/14 Lecture 15: H Control Synthesis Example Sub-optimal H Controllers

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 2/14 Example Consider a control system ẋ = x + w, y = 2x w, z = x u

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 2/14 Example Consider a control system ẋ = x + w, y = 2x w, z = x u Find a controller that minimizes its H -norm from w to z, i.e. Find a set of internally stabilizing controllers u = K(s)y Determine the one that ensures the inequality + 0 z(t) 2 dt γ 2 + 0 w(t) 2 dt, w L 2 (R + ) with minimal possible constant γ > 0.

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 3/14 Solution: The system is evolving in open loop; so, the controller is stabilizing if it is stable!

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 3/14 Solution: The system is evolving in open loop; so, the controller is stabilizing if it is stable! The transfer function from w to z is defined by [ 1 z = T w z (s) w = s + 1 K(s)1 s ] 1 + s w

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 3/14 Solution: The system is evolving in open loop; so, the controller is stabilizing if it is stable! The transfer function from w to z is defined by [ 1 z = T w z (s) w = s + 1 K(s)1 s ] 1 + s w The inequality that defines γ + 0 + z(t) 2 dt γ 2 0 can be rewritten as follows with w(jω) = w(t) 2 dt, w L 2 (R + ) + 0 e jωt w(t)dt 1 2π + T w z (jω) w(jω) 2 dω γ 2 + 2π 1 w(jω) 2 dω

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 3/14 Solution: The system is evolving in open loop; so, the controller is stabilizing if it is stable! The transfer function from w to z is defined by [ 1 z = T w z (s) w = s + 1 K(s)1 s ] 1 + s w The inequality that defines γ + 0 + z(t) 2 dt γ 2 0 becomes the condition γ = sup ω R T w z (jω) = sup s:res 0 w(t) 2 dt, w L 2 (R + ) 1 s + 1 K(s)1 s 1 + s

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 3/14 Solution: The system is evolving in open loop; so, the controller is stabilizing if it is stable! The transfer function from w to z is defined by [ 1 z = T w z (s) w = s + 1 K(s)1 s ] 1 + s w The inequality that defines γ + 0 + z(t) 2 dt γ 2 0 w(t) 2 dt, w L 2 (R + ) becomes the condition γ = sup 1 s:res 0 s + 1 K(s)1 s 1 + s... s=1 = 1 2

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 4/14 Example Consider a control system ẋ = x + w, y = 2x w, z = x u We have found that the H norm of the system is not less than 1 2 ; It is easy to see that one of optimal controllers is K(s) = 1 2 ; The corresponding transfer function T w z (s) is equal to 1 2, that is all-pass

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 4/14 Example Consider a control system ẋ = x + w, y = 2x w, z = x u We have found that the H norm of the system is not less than 1 2 ; It is easy to see that one of optimal controllers is K(s) = 1 2 ; The corresponding transfer function T w z (s) is equal to 1 2, that is all-pass This property makes this optimal controller less attractive, and motivates the search of a sub-optimal feedback.

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 5/14 Important Statement The following two statements are equivalent:

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 5/14 Important Statement The following two statements are equivalent: The matrix A is Hurwitz matrix, and has H -norm less than γ G(s) = D + C ( si n A ) 1 B

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 5/14 Important Statement The following two statements are equivalent: The matrix A is Hurwitz matrix, and has H -norm less than γ G(s) = D + C ( si n A ) 1 B There exists P = P T > 0 such that the quadratic form σ P (x,w) = γ 2 w 2 Cx + Dw 2 2xT P ( Ax + Bw ) is strictly positive definite

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 6/14 Simplified Sub-Optimal Control Design Consider a control system [ ] [ d dt x = Ax + B (1) 1, w 1 B(2) 1 z = C(1) 1 C (2) 1 y = C 2 x + [ x + D(1) 12 D (2) 12 D (1) 21, D(2) 21 w 2 u ] ] [ w 1 w 2 + B 2 u ]

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 6/14 Simplified Sub-Optimal Control Design Consider a control system [ ] [ d dt x = Ax + B (1) 1, 0 w 1 z = C(1) 1 0 y = C 2 x + [ x + 0, I 0 I w 2 u ] [ w 1 w 2 ] ] + B 2 u

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 6/14 Simplified Sub-Optimal Control Design Consider a control system d dt x = Ax + B 1w 1 + B 2 u z = C 1 x + 0 u 0 I y = C 2 x + w 2

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 6/14 Simplified Sub-Optimal Control Design Consider a control system d dt x = Ax + B 1w 1 + B 2 u z = C 1 x + 0 u 0 I y = C 2 x + w 2 The problem is to design a controller d dt x c = A c x c + B c y, u = C c x c + D c y, such that the closed-loop system is internally stable and Tw z (s) γ

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 7/14 Solution: Combining the plant and the controller, we obtain the closed-loop system d dt [ x x c ] = z = [ ] (A + B 2 D c C 2 ) B 2 C c } B c C 2 {{ A c } = A cl C [ ] 1 0 x + D c C 2 C c x c } {{ } = C cl [ x x c ] 0 0 0 D c }{{} D cl + [ ] B 1 B 2 D c 0 B c }{{} = B cl [ ] w 1 w 2 [ ] w 1 w 2

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 7/14 Solution: Combining the plant and the controller, we obtain the closed-loop system d dt [ x x c ] = z = [ ] (A + B 2 D c C 2 ) B 2 C c } B c C 2 {{ A c } = A cl C [ ] 1 0 x + D c C 2 C c x c } {{ } = C cl [ x x c ] 0 0 0 D c }{{} D cl + [ ] B 1 B 2 D c 0 B c }{{} = B cl [ ] w 1 w 2 [ ] w 1 w 2 The transfer function from w to z is then T w z (s) = D cl + C cl ( si Acl ) 1Bcl

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 8/14 Important Statement The following two statements are equivalent:

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 8/14 Important Statement The following two statements are equivalent: The matrix A is Hurwitz matrix, and has H -norm less than γ G(s) = D + C ( si n A ) 1 B

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 8/14 Important Statement The following two statements are equivalent: The matrix A is Hurwitz matrix, and has H -norm less than γ G(s) = D + C ( si n A ) 1 B There exists P = P T > 0 such that the quadratic form σ P (x,w) = γ 2 w 2 Cx + Dw 2 2xT P ( Ax + Bw ) is strictly positive definite

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 9/14 Solution: For our case G(s) = D cl + C cl ( si Acl ) 1Bcl and the condition σ P ( ) > 0 is σ P ([x T,x T c ],[wt 1,wT 2 ]) = γ2 w 2 Ccl X + D cl w 2 2X T P ( A cl X + B cl w )

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 9/14 Solution: For our case G(s) = D cl + C cl ( si Acl ) 1Bcl and the condition σ P ( ) > 0 is σ P ([x T,x T c ],[wt 1,wT 2 ]) = γ2 w 2 Ccl X + D cl w 2 2X T P ( A cl X + B cl w ) ( w1 = γ 2 2 ) + w2 2 C1 ( x 2 + ) u 2 [ ][ ][ ] x P 11 P 12 Ax + B 1 w 1 + B 2 u 2 P12 T P 22 A c x c + B c y x c > 0, w 1, w 2, x, x c 0

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 10/14 Solution: It turns out that the condition σ P ( ) > 0 is valid if: There is X > 0 such that C T 1 C 1 + XA + A T X = X ( B 2 B2 T 1 ) γ 2B 1B1 T X

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 10/14 Solution: It turns out that the condition σ P ( ) > 0 is valid if: There is X > 0 such that C T 1 C 1 + XA + A T X = X ( B 2 B2 T 1 ) γ 2B 1B1 T X There is Y > 0 such that B T 1 B 1 + Y A T + AY = Y ( C 2 C2 T 1 ) γ 2C 1C1 T Y

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 10/14 Solution: It turns out that the condition σ P ( ) > 0 is valid if: There is X > 0 such that C T 1 C 1 + XA + A T X = X ( B 2 B2 T 1 ) γ 2B 1B1 T X There is Y > 0 such that B T 1 B 1 + Y A T + AY = Y ( C 2 C2 T 1 ) γ 2C 1C1 T Y γ 2 Y 1 > X

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 11/14 Solution: One of suboptimal controllers is then u = B T 2 Xˆx d dtˆx = Aˆx + B 1ŵ 1 + B 2 u + L 1 (C 2ˆx y) ŵ 1 = 1 γ 2BT 1 Xˆx where L 1 = ( I 1 γ 2Y X ) 1 Y C T 2

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 12/14 Modified Problem 13.2 Consider a one degree of freedom control loop with the plant P(s) = s 10 ( s + 10 ) (s + 1), and weighting transfer functions W e (s) = 1 s + 0.001, W u(s) = s + 2 s + 10

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 12/14 Modified Problem 13.2 Consider a one degree of freedom control loop with the plant P(s) = s 10 ( s + 10 ) (s + 1), and weighting transfer functions W e (s) = 1 s + 0.001, W u(s) = s + 2 s + 10 Design a controller K(s) that minimizes W e ( )S( ), S(s) = W u ( )K( )S( ) 1 1 + P(s)K(s) Simulate a step response of the closed-loop system

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 13/14 Solution of Modified Problem 13.2 Last lecture we have rewritten the problem for H 2 -minimization in the standard form: [ ] [ ] [ ] w z w ẋ = Ax + B, = C x + D u y u defining the generalized plant with G(s) = C (si A) 1 B + D, dimy = 1, dimu = 1

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 13/14 Solution of Modified Problem 13.2 Last lecture we have rewritten the problem for H 2 -minimization in the standard form: [ ] [ ] [ ] w z w ẋ = Ax + B, = C x + D u y u defining the generalized plant with G(s) = C (si A) 1 B + D, dimy = 1, dimu = 1 H 2 optimal regulator can be obtained in Matlab using: h2syn(g,1,1)

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 13/14 Solution of Modified Problem 13.2 Last lecture we have rewritten the problem for H 2 -minimization in the standard form: [ ] [ ] [ ] w z w ẋ = Ax + B, = C x + D u y u defining the generalized plant with G(s) = C (si A) 1 B + D, dimy = 1, dimu = 1 H 2 optimal regulator can be obtained in Matlab using: h2syn(g,1,1) H optimal regulator can be obtained in Matlab using: hinfsyn(g,1,1)

c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 14/14 Final Week Presentations of the final projects: and March 16, 10:15-12:00, in A206 March 17, 10:15-12:00, in A205 Final exam: March 19, 13:00-16:00, in A206

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