k -Step Ahead Prediction Error Model 1. k -Step Ahead Prediction - PowerPoint PPT Presentation

Example: Splitting Noise into Past and Future 5.

Example: Splitting Noise into Past and Future 5. 1 + 0 . 5 z − 1 u ( n + j − 2) y ( n + j ) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 + 1 − 0 . 6 z − 1 − 0 . 16 z − 2 ξ ( n + j )

Example: Splitting Noise into Past and Future 5. 1 + 0 . 5 z − 1 u ( n + j − 2) y ( n + j ) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 + 1 − 0 . 6 z − 1 − 0 . 16 z − 2 ξ ( n + j ) Split C into E j and F j , for j = 2 :

Example: Splitting Noise into Past and Future 5. 1 + 0 . 5 z − 1 u ( n + j − 2) y ( n + j ) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 + 1 − 0 . 6 z − 1 − 0 . 16 z − 2 ξ ( n + j ) Split C into E j and F j , for j = 2 : 1 + 0 . 5 z − 1 0 . 82 + 0 . 176 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 = (1 + 1 . 1 z − 1 ) + z − 2 1 − 0 . 6 z − 1 − 0 . 16 z − 2

Example: Splitting Noise into Past and Future 5. 1 + 0 . 5 z − 1 u ( n + j − 2) y ( n + j ) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 + 1 − 0 . 6 z − 1 − 0 . 16 z − 2 ξ ( n + j ) Split C into E j and F j , for j = 2 : 1 + 0 . 5 z − 1 0 . 82 + 0 . 176 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 = (1 + 1 . 1 z − 1 ) + z − 2 1 − 0 . 6 z − 1 − 0 . 16 z − 2 Substitute it in the expression for y ( n + j ) ,

Example: Splitting Noise into Past and Future 5. 1 + 0 . 5 z − 1 u ( n + j − 2) y ( n + j ) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 + 1 − 0 . 6 z − 1 − 0 . 16 z − 2 ξ ( n + j ) Split C into E j and F j , for j = 2 : 1 + 0 . 5 z − 1 0 . 82 + 0 . 176 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 = (1 + 1 . 1 z − 1 ) + z − 2 1 − 0 . 6 z − 1 − 0 . 16 z − 2 Substitute it in the expression for y ( n + j ) , with j = 2 :

Example: Splitting Noise into Past and Future 5. 1 + 0 . 5 z − 1 u ( n + j − 2) y ( n + j ) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 + 1 − 0 . 6 z − 1 − 0 . 16 z − 2 ξ ( n + j ) Split C into E j and F j , for j = 2 : 1 + 0 . 5 z − 1 0 . 82 + 0 . 176 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 = (1 + 1 . 1 z − 1 ) + z − 2 1 − 0 . 6 z − 1 − 0 . 16 z − 2 Substitute it in the expression for y ( n + j ) , with j = 2 : 1 y ( n + 2) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 u ( n )

Example: Splitting Noise into Past and Future 5. 1 + 0 . 5 z − 1 u ( n + j − 2) y ( n + j ) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 + 1 − 0 . 6 z − 1 − 0 . 16 z − 2 ξ ( n + j ) Split C into E j and F j , for j = 2 : 1 + 0 . 5 z − 1 0 . 82 + 0 . 176 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 = (1 + 1 . 1 z − 1 ) + z − 2 1 − 0 . 6 z − 1 − 0 . 16 z − 2 Substitute it in the expression for y ( n + j ) , with j = 2 : 1 y ( n + 2) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 u ( n ) + (1 + 1 . 1 z − 1 ) ξ ( n + 2)

Example: Splitting Noise into Past and Future 5. 1 + 0 . 5 z − 1 u ( n + j − 2) y ( n + j ) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 + 1 − 0 . 6 z − 1 − 0 . 16 z − 2 ξ ( n + j ) Split C into E j and F j , for j = 2 : 1 + 0 . 5 z − 1 0 . 82 + 0 . 176 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 = (1 + 1 . 1 z − 1 ) + z − 2 1 − 0 . 6 z − 1 − 0 . 16 z − 2 Substitute it in the expression for y ( n + j ) , with j = 2 : 1 y ( n + 2) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 u ( n ) + (1 + 1 . 1 z − 1 ) ξ ( n + 2) 0 . 82 + 0 . 176 z − 1 + z − 2 1 − 0 . 6 z − 1 − 0 . 16 z − 2 ξ ( n + 2)

Example: Splitting Noise into Past and Future 5. 1 + 0 . 5 z − 1 u ( n + j − 2) y ( n + j ) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 + 1 − 0 . 6 z − 1 − 0 . 16 z − 2 ξ ( n + j ) Split C into E j and F j , for j = 2 : 1 + 0 . 5 z − 1 0 . 82 + 0 . 176 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 = (1 + 1 . 1 z − 1 ) + z − 2 1 − 0 . 6 z − 1 − 0 . 16 z − 2 Substitute it in the expression for y ( n + j ) , with j = 2 : 1 y ( n + 2) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 u ( n ) + (1 + 1 . 1 z − 1 ) ξ ( n + 2) 0 . 82 + 0 . 176 z − 1 + z − 2 1 − 0 . 6 z − 1 − 0 . 16 z − 2 ξ ( n + 2) Second term is unknown;

Example: Splitting Noise into Past and Future 5. 1 + 0 . 5 z − 1 u ( n + j − 2) y ( n + j ) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 + 1 − 0 . 6 z − 1 − 0 . 16 z − 2 ξ ( n + j ) Split C into E j and F j , for j = 2 : 1 + 0 . 5 z − 1 0 . 82 + 0 . 176 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 = (1 + 1 . 1 z − 1 ) + z − 2 1 − 0 . 6 z − 1 − 0 . 16 z − 2 Substitute it in the expression for y ( n + j ) , with j = 2 : 1 y ( n + 2) = 1 − 0 . 6 z − 1 − 0 . 16 z − 2 u ( n ) + (1 + 1 . 1 z − 1 ) ξ ( n + 2) 0 . 82 + 0 . 176 z − 1 + z − 2 1 − 0 . 6 z − 1 − 0 . 16 z − 2 ξ ( n + 2) Second term is unknown; Last term is known. 5 Digital Control Kannan M. Moudgalya, Autumn 2007

Splitting Noise into Past and Future 6.

Splitting Noise into Past and Future 6. Ay ( n ) = Bu ( n − k ) + Cξ ( n )

Splitting Noise into Past and Future 6. Ay ( n ) = Bu ( n − k ) + Cξ ( n ) y ( n + j ) = B Au ( n + j − k ) + C Aξ ( n + j )

Splitting Noise into Past and Future 6. Ay ( n ) = Bu ( n − k ) + Cξ ( n ) y ( n + j ) = B Au ( n + j − k ) + C Aξ ( n + j ) = B E j + z − j F j � � Au ( n + j − k ) + ξ ( n + j ) A

Splitting Noise into Past and Future 6. Ay ( n ) = Bu ( n − k ) + Cξ ( n ) y ( n + j ) = B Au ( n + j − k ) + C Aξ ( n + j ) = B E j + z − j F j � � Au ( n + j − k ) + ξ ( n + j ) A = B Au ( n + j − k ) + F j A ξ ( n ) + E j ξ ( n + j )

Splitting Noise into Past and Future 6. Ay ( n ) = Bu ( n − k ) + Cξ ( n ) y ( n + j ) = B Au ( n + j − k ) + C Aξ ( n + j ) = B E j + z − j F j � � Au ( n + j − k ) + ξ ( n + j ) A = B Au ( n + j − k ) + F j A ξ ( n ) + E j ξ ( n + j ) = B Au ( n + j − k ) + F j Ay ( n ) − Bu ( n − k ) + E j ξ ( n + j ) A C

Splitting Noise into Past and Future 6. Ay ( n ) = Bu ( n − k ) + Cξ ( n ) y ( n + j ) = B Au ( n + j − k ) + C Aξ ( n + j ) = B E j + z − j F j � � Au ( n + j − k ) + ξ ( n + j ) A = B Au ( n + j − k ) + F j A ξ ( n ) + E j ξ ( n + j ) = B Au ( n + j − k ) + F j Ay ( n ) − Bu ( n − k ) + E j ξ ( n + j ) A C = B Au ( n + j − k ) − F j B AC u ( n − k ) + F j C y ( n ) + E j ξ ( n + j )

Splitting Noise into Past and Future 6. Ay ( n ) = Bu ( n − k ) + Cξ ( n ) y ( n + j ) = B Au ( n + j − k ) + C Aξ ( n + j ) = B E j + z − j F j � � Au ( n + j − k ) + ξ ( n + j ) A = B Au ( n + j − k ) + F j A ξ ( n ) + E j ξ ( n + j ) = B Au ( n + j − k ) + F j Ay ( n ) − Bu ( n − k ) + E j ξ ( n + j ) A C = B Au ( n + j − k ) − F j B AC u ( n − k ) + F j C y ( n ) + E j ξ ( n + j ) = B 1 − F j u ( n + j − k ) + F j � � C z − j C y ( n ) + E j ξ ( n + j ) A 6 Digital Control Kannan M. Moudgalya, Autumn 2007

Splitting Noise into Past and Future 7.

Splitting Noise into Past and Future 7. From the previous slide, y ( n + j ) = B � 1 − F j � u ( n + j − k ) + F j C z − j C y ( n ) + E j ξ ( n + j ) A

Splitting Noise into Past and Future 7. From the previous slide, y ( n + j ) = B � 1 − F j � u ( n + j − k ) + F j C z − j C y ( n ) + E j ξ ( n + j ) A C A = E j + z − j F j A

Splitting Noise into Past and Future 7. From the previous slide, y ( n + j ) = B � 1 − F j � u ( n + j − k ) + F j C z − j C y ( n ) + E j ξ ( n + j ) A C A = E j + z − j F j A ⇒ C A − z − j F j A = E j

Splitting Noise into Past and Future 7. From the previous slide, y ( n + j ) = B � 1 − F j � u ( n + j − k ) + F j C z − j C y ( n ) + E j ξ ( n + j ) A C A = E j + z − j F j A ⇒ C A − z − j F j A = E j ⇒ C 1 − z − j F j � � = E j A C

Splitting Noise into Past and Future 7. From the previous slide, y ( n + j ) = B � 1 − F j � u ( n + j − k ) + F j C z − j C y ( n ) + E j ξ ( n + j ) A C A = E j + z − j F j A ⇒ C A − z − j F j A = E j ⇒ C 1 − z − j F j � � = E j A C y ( n + j ) = E j B C u ( n + j − k ) + F j C y ( n ) + E j ξ ( n + j )

Splitting Noise into Past and Future 7. From the previous slide, y ( n + j ) = B � 1 − F j � u ( n + j − k ) + F j C z − j C y ( n ) + E j ξ ( n + j ) A C A = E j + z − j F j A ⇒ C A − z − j F j A = E j ⇒ C 1 − z − j F j � � = E j A C y ( n + j ) = E j B C u ( n + j − k ) + F j C y ( n ) + E j ξ ( n + j ) Last term has only future terms.

Splitting Noise into Past and Future 7. From the previous slide, y ( n + j ) = B � 1 − F j � u ( n + j − k ) + F j C z − j C y ( n ) + E j ξ ( n + j ) A C A = E j + z − j F j A ⇒ C A − z − j F j A = E j ⇒ C 1 − z − j F j � � = E j A C y ( n + j ) = E j B C u ( n + j − k ) + F j C y ( n ) + E j ξ ( n + j ) Last term has only future terms. Hence, best prediction model:

Splitting Noise into Past and Future 7. From the previous slide, y ( n + j ) = B � 1 − F j � u ( n + j − k ) + F j C z − j C y ( n ) + E j ξ ( n + j ) A C A = E j + z − j F j A ⇒ C A − z − j F j A = E j ⇒ C 1 − z − j F j � � = E j A C y ( n + j ) = E j B C u ( n + j − k ) + F j C y ( n ) + E j ξ ( n + j ) Last term has only future terms. Hence, best prediction model: y ( n + j | n ) = E j B C u ( n + j − k ) + F j ˆ C y ( n )

Splitting Noise into Past and Future 7. From the previous slide, y ( n + j ) = B � 1 − F j � u ( n + j − k ) + F j C z − j C y ( n ) + E j ξ ( n + j ) A C A = E j + z − j F j A ⇒ C A − z − j F j A = E j ⇒ C 1 − z − j F j � � = E j A C y ( n + j ) = E j B C u ( n + j − k ) + F j C y ( n ) + E j ξ ( n + j ) Last term has only future terms. Hence, best prediction model: y ( n + j | n ) = E j B C u ( n + j − k ) + F j ˆ C y ( n ) ˆ means estimate.

Splitting Noise into Past and Future 7. From the previous slide, y ( n + j ) = B � 1 − F j � u ( n + j − k ) + F j C z − j C y ( n ) + E j ξ ( n + j ) A C A = E j + z − j F j A ⇒ C A − z − j F j A = E j ⇒ C 1 − z − j F j � � = E j A C y ( n + j ) = E j B C u ( n + j − k ) + F j C y ( n ) + E j ξ ( n + j ) Last term has only future terms. Hence, best prediction model: y ( n + j | n ) = E j B C u ( n + j − k ) + F j ˆ C y ( n ) ˆ means estimate. | n means “using measurements, available up to and including n ”. 7 Digital Control Kannan M. Moudgalya, Autumn 2007

Example: Splitting C/A into E j and F j 8.

Example: Splitting C/A into E j and F j 8. 1 + 0 . 5 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 = C A

Example: Splitting C/A into E j and F j 8. 1 + 0 . 5 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 = C A = E j + z − j F j A

Example: Splitting C/A into E j and F j 8. 1 + 0 . 5 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 = C A = E j + z − j F j A 1 + 1 . 1 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 | 1 +0 . 5 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 +1 . 1 z − 1 +0 . 16 z − 2 +1 . 1 z − 1 − 0 . 66 z − 2 − 0 . 176 z − 3 +0 . 82 z − 2 +0 . 176 z − 3

Example: Splitting C/A into E j and F j 8. 1 + 0 . 5 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 = C A = E j + z − j F j A 1 + 1 . 1 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 | 1 +0 . 5 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 +1 . 1 z − 1 +0 . 16 z − 2 +1 . 1 z − 1 − 0 . 66 z − 2 − 0 . 176 z − 3 +0 . 82 z − 2 +0 . 176 z − 3 1 + 0 . 5 z − 1 0 . 82 + 0 . 176 z − 1 1 − 0 . 6 z − 1 − 0 . 16 z − 2 = (1 + 1 . 1 z − 1 ) + z − 2 1 − 0 . 6 z − 1 − 0 . 16 z − 2 8 Digital Control Kannan M. Moudgalya, Autumn 2007

Another Method to Split C/A into E j and F j 9.

Another Method to Split C/A into E j and F j 9. An easier method exists to solve C A = E j + z − j F j A

Another Method to Split C/A into E j and F j 9. An easier method exists to solve C A = E j + z − j F j A Cross multiply by A :

Another Method to Split C/A into E j and F j 9. An easier method exists to solve C A = E j + z − j F j A Cross multiply by A : C = AE j + z − j F j

Another Method to Split C/A into E j and F j 9. An easier method exists to solve C A = E j + z − j F j A Cross multiply by A : C = AE j + z − j F j • C , A , z − j are known

Another Method to Split C/A into E j and F j 9. An easier method exists to solve C A = E j + z − j F j A Cross multiply by A : C = AE j + z − j F j • C , A , z − j are known • E j , F j are to be calculated.

Another Method to Split C/A into E j and F j 9. An easier method exists to solve C A = E j + z − j F j A Cross multiply by A : C = AE j + z − j F j • C , A , z − j are known • E j , F j are to be calculated. • Think: How would you solve it? 9 Digital Control Kannan M. Moudgalya, Autumn 2007

Different Noise and Prediction Models: AR- 10. MAX

Different Noise and Prediction Models: AR- 10. MAX ARMAX Model

Different Noise and Prediction Models: AR- 10. MAX ARMAX Model : Ay ( n ) = Bu ( n − k ) + Cξ ( n )

Different Noise and Prediction Models: AR- 10. MAX ARMAX Model : Ay ( n ) = Bu ( n − k ) + Cξ ( n ) C = E j A + z − j F j

Different Noise and Prediction Models: AR- 10. MAX ARMAX Model : Ay ( n ) = Bu ( n − k ) + Cξ ( n ) C = E j A + z − j F j y ( n + j | t ) = E j B C u ( n + j − k ) + F j ˆ C y ( n ) 10 Digital Control Kannan M. Moudgalya, Autumn 2007

Different Noise and Prediction Models: ARI- 11. MAX

Different Noise and Prediction Models: ARI- 11. MAX ARIMAX model

Different Noise and Prediction Models: ARI- 11. MAX ARIMAX model with ∆ = 1 − z − 1 :

Different Noise and Prediction Models: ARI- 11. MAX ARIMAX model with ∆ = 1 − z − 1 : Ay ( n ) = Bu ( n − k ) + C ∆ ξ ( n )

Different Noise and Prediction Models: ARI- 11. MAX ARIMAX model with ∆ = 1 − z − 1 : Ay ( n ) = Bu ( n − k ) + C ∆ ξ ( n ) A ∆ y ( n ) = B ∆ u ( n − k ) + Cξ ( n )

Different Noise and Prediction Models: ARI- 11. MAX ARIMAX model with ∆ = 1 − z − 1 : Ay ( n ) = Bu ( n − k ) + C ∆ ξ ( n ) A ∆ y ( n ) = B ∆ u ( n − k ) + Cξ ( n ) Recall ARMAX model: Ay ( n ) = Bu ( n − k ) + Cξ ( n )

Different Noise and Prediction Models: ARI- 11. MAX ARIMAX model with ∆ = 1 − z − 1 : Ay ( n ) = Bu ( n − k ) + C ∆ ξ ( n ) A ∆ y ( n ) = B ∆ u ( n − k ) + Cξ ( n ) Recall ARMAX model: Ay ( n ) = Bu ( n − k ) + Cξ ( n ) Is the solution for ARMAX model useful?

Different Noise and Prediction Models: ARI- 11. MAX ARIMAX model with ∆ = 1 − z − 1 : Ay ( n ) = Bu ( n − k ) + C ∆ ξ ( n ) A ∆ y ( n ) = B ∆ u ( n − k ) + Cξ ( n ) Recall ARMAX model: Ay ( n ) = Bu ( n − k ) + Cξ ( n ) Is the solution for ARMAX model useful? A ← A ∆

Different Noise and Prediction Models: ARI- 11. MAX ARIMAX model with ∆ = 1 − z − 1 : Ay ( n ) = Bu ( n − k ) + C ∆ ξ ( n ) A ∆ y ( n ) = B ∆ u ( n − k ) + Cξ ( n ) Recall ARMAX model: Ay ( n ) = Bu ( n − k ) + Cξ ( n ) Is the solution for ARMAX model useful? A ← A ∆ , B ← B ∆

Different Noise and Prediction Models: ARI- 11. MAX ARIMAX model with ∆ = 1 − z − 1 : Ay ( n ) = Bu ( n − k ) + C ∆ ξ ( n ) A ∆ y ( n ) = B ∆ u ( n − k ) + Cξ ( n ) Recall ARMAX model: Ay ( n ) = Bu ( n − k ) + Cξ ( n ) Is the solution for ARMAX model useful? A ← A ∆ , B ← B ∆ C = E j A ∆ + z − j F j

Different Noise and Prediction Models: ARI- 11. MAX ARIMAX model with ∆ = 1 − z − 1 : Ay ( n ) = Bu ( n − k ) + C ∆ ξ ( n ) A ∆ y ( n ) = B ∆ u ( n − k ) + Cξ ( n ) Recall ARMAX model: Ay ( n ) = Bu ( n − k ) + Cξ ( n ) Is the solution for ARMAX model useful? A ← A ∆ , B ← B ∆ C = E j A ∆ + z − j F j y ( n + j | n ) = E j B ∆ u ( n + j − k ) + F j ˆ C y ( n ) C 11 Digital Control Kannan M. Moudgalya, Autumn 2007

Different Noise and Prediction Models: ARIX 12.

Different Noise and Prediction Models: ARIX 12. Recall ARIMAX model from previous slide: A ∆ y ( n ) = B ∆ u ( n − k ) + Cξ ( n ) y ( n + j | n ) = E j B ∆ u ( n + j − k ) + F j ˆ C y ( n ) C

Different Noise and Prediction Models: ARIX 12. Recall ARIMAX model from previous slide: A ∆ y ( n ) = B ∆ u ( n − k ) + Cξ ( n ) y ( n + j | n ) = E j B ∆ u ( n + j − k ) + F j ˆ C y ( n ) C ARIX model,

Different Noise and Prediction Models: ARIX 12. Recall ARIMAX model from previous slide: A ∆ y ( n ) = B ∆ u ( n − k ) + Cξ ( n ) y ( n + j | n ) = E j B ∆ u ( n + j − k ) + F j ˆ C y ( n ) C ARIX model, obtained with C = 1 in ARIMAX:

Different Noise and Prediction Models: ARIX 12. Recall ARIMAX model from previous slide: A ∆ y ( n ) = B ∆ u ( n − k ) + Cξ ( n ) y ( n + j | n ) = E j B ∆ u ( n + j − k ) + F j ˆ C y ( n ) C ARIX model, obtained with C = 1 in ARIMAX: Ay ( n ) = Bu ( n − k ) + 1 ∆ ξ ( n )

Different Noise and Prediction Models: ARIX 12. Recall ARIMAX model from previous slide: A ∆ y ( n ) = B ∆ u ( n − k ) + Cξ ( n ) y ( n + j | n ) = E j B ∆ u ( n + j − k ) + F j ˆ C y ( n ) C ARIX model, obtained with C = 1 in ARIMAX: Ay ( n ) = Bu ( n − k ) + 1 ∆ ξ ( n ) 1 = E j A ∆ + z − j F j

Different Noise and Prediction Models: ARIX 12. Recall ARIMAX model from previous slide: A ∆ y ( n ) = B ∆ u ( n − k ) + Cξ ( n ) y ( n + j | n ) = E j B ∆ u ( n + j − k ) + F j ˆ C y ( n ) C ARIX model, obtained with C = 1 in ARIMAX: Ay ( n ) = Bu ( n − k ) + 1 ∆ ξ ( n ) 1 = E j A ∆ + z − j F j y ( n + j | t ) = E j B ∆ u ( n + j − k ) + F j y ( n ) ˆ 12 Digital Control Kannan M. Moudgalya, Autumn 2007

Minimum Variance Control: Regulation 13.

Minimum Variance Control: Regulation 13. ARMAX Model: Ay ( n ) = Bu ( n − k ) + Cξ ( n )