Preliminary Results on Generalized Fixed Order Interpolation - PowerPoint PPT Presentation

Preliminary Results on Generalized Fixed Order Interpolation Constantino Lagoa 1 1 Electrical Engineering Department The Pennsylvania State University USA Workshop on Uncertain Dynamical Systems 2011 Work done in collaboration with Chao Feng and Mario Sznaier Lagoa (Penn State) Fixed Order Interpolation Udine 2011 1 / 27

A General Interpolation Problem Given Set of pairs of complex numbers F . = { ( z 0 , F 0 ) , ( z 1 , F 1 ) , . . . , ( z N NP , F N NP ) } , Set of real values T . = { g 0 , g 1 , . . . , g N CF } Supply function s ( u , y ) Bound on system order N . Interpolation Problem Find a causal SISO system G ( z ) of order no greater N satisfying G ( z k ) = F k , k = 0 , . . . , N NP ; G ( z ) = g 0 + g 1 z − 1 + . . . + g N CF z − N CF + . . . ; G ( z ) is passive with given supply function s ( u , y ) . Lagoa (Penn State) Fixed Order Interpolation Udine 2011 2 / 27

A General Interpolation Problem Given Set of pairs of complex numbers F . = { ( z 0 , F 0 ) , ( z 1 , F 1 ) , . . . , ( z N NP , F N NP ) } , Set of real values T . = { g 0 , g 1 , . . . , g N CF } Supply function s ( u , y ) Bound on system order N . Interpolation Problem Find a causal SISO system G ( z ) of order no greater N satisfying G ( z k ) = F k , k = 0 , . . . , N NP ; G ( z ) = g 0 + g 1 z − 1 + . . . + g N CF z − N CF + . . . ; G ( z ) is passive with given supply function s ( u , y ) . Lagoa (Penn State) Fixed Order Interpolation Udine 2011 2 / 27

A General Interpolation Problem Given Set of pairs of complex numbers F . = { ( z 0 , F 0 ) , ( z 1 , F 1 ) , . . . , ( z N NP , F N NP ) } , Set of real values T . = { g 0 , g 1 , . . . , g N CF } Supply function s ( u , y ) Bound on system order N . Interpolation Problem Find a causal SISO system G ( z ) of order no greater N satisfying G ( z k ) = F k , k = 0 , . . . , N NP ; G ( z ) = g 0 + g 1 z − 1 + . . . + g N CF z − N CF + . . . ; G ( z ) is passive with given supply function s ( u , y ) . Lagoa (Penn State) Fixed Order Interpolation Udine 2011 2 / 27

A General Interpolation Problem Given Set of pairs of complex numbers F . = { ( z 0 , F 0 ) , ( z 1 , F 1 ) , . . . , ( z N NP , F N NP ) } , Set of real values T . = { g 0 , g 1 , . . . , g N CF } Supply function s ( u , y ) Bound on system order N . Interpolation Problem Find a causal SISO system G ( z ) of order no greater N satisfying G ( z k ) = F k , k = 0 , . . . , N NP ; G ( z ) = g 0 + g 1 z − 1 + . . . + g N CF z − N CF + . . . ; G ( z ) is passive with given supply function s ( u , y ) . Lagoa (Penn State) Fixed Order Interpolation Udine 2011 2 / 27

Application to Control Systems This is a general framework that includes many problems in the control area Fixed order H ∞ controller design. Simultaneous stabilization. Fixed order system system identification. Spectral estimation. Lagoa (Penn State) Fixed Order Interpolation Udine 2011 3 / 27

Related Work Nevalinna-Pick interpolation Caratheodory-Fejer interpolation Mixed-domain system identification Polynomial approaches to fixed-order system identification Many more..... Lagoa (Penn State) Fixed Order Interpolation Udine 2011 4 / 27

Our Contribution The general interpolation problem formulated here is a very complex problem. Objective Develop computationally efficient relaxations of this problem by Reformulating the problem as finding a point in a semialgebraic set Using results from the areas of polynomial optimization and sparsification Lagoa (Penn State) Fixed Order Interpolation Udine 2011 5 / 27

Our Contribution The general interpolation problem formulated here is a very complex problem. Objective Develop computationally efficient relaxations of this problem by Reformulating the problem as finding a point in a semialgebraic set Using results from the areas of polynomial optimization and sparsification Lagoa (Penn State) Fixed Order Interpolation Udine 2011 5 / 27

Setup Fixed order plant G ( z ) = b 0 + b 1 z − 1 + · · · + b m z − m 1 + a 1 z − 1 + · · · + a n z − n Design parameters Θ . = [ − a 1 , . . . , − a n , b 0 , . . . , b m ] T Supply function � � π 11 � � u � s ( u , y ) . π 12 � = u y π 12 π 22 y Lagoa (Penn State) Fixed Order Interpolation Udine 2011 6 / 27

Consistency Set as a Semialgebraic Set Time Domain Interpolation Time domain interpolation conditions are linear equalities on the decision variables. Algebraic Description The time domain interpolation conditions are satisfied if and only if k (Θ) . = Θ T · φ k − g k = 0 , p CF k = 0 , . . . , N CF where φ k . = [ g k − 1 , . . . , g k − n , u k , . . . , u k − m ] T is the regressor and u k is the impulse function. Lagoa (Penn State) Fixed Order Interpolation Udine 2011 7 / 27

Consistency Set as a Semialgebraic Set Frequency Domain Interpolation Frequency domain conditions are also linear equalities on the decision variables. Algebraic Description The frequency domain interpolation conditions are satisfied if and only if . p NP 2 k (Θ) = Re { b ( z k ) − F k a ( z k ) } = 0 , . p NP 2 k + 1 (Θ) = Im { b ( z k ) − F k a ( z k ) } = 0 for k = 0 , . . . , N NP . Lagoa (Penn State) Fixed Order Interpolation Udine 2011 8 / 27

Consistency Set as a Semialgebraic Set Stability - Jury’s Criterion Jury’s array a n a n − 1 a n − 2 · · · a 2 a 1 1 1 a 1 a 2 · · · a n − 2 a n − 1 a n J 1 , n − 1 J 1 , n − 2 J 1 , n − 3 · · · J 1 , 1 J 1 , 0 J 1 , 0 J 1 , 1 J 1 , 2 · · · J 1 , n − 2 J 1 , n − 1 J 2 , n − 2 J 2 , n − 3 J 2 , n − 4 · · · J 2 , 0 J 2 , 0 J 2 , 1 J 2 , 2 · · · J 2 , n − 2 . . . . . . . . . J n − 2 , 2 J n − 2 , 1 J n − 2 , 0 � � J k , i . J k − 1 , n − k J k − 1 , i � � = � , � � 1 J k − 1 , n − k − i � Lagoa (Penn State) Fixed Order Interpolation Udine 2011 9 / 27

Consistency Set as a Semialgebraic Set Stability - Jury’s Criterion Lemma The system G ( z ) is stable, i.e., all the roots of a ( z ) locate inside the unit circle, if and only if the following inequalities hold, n � 1 + a i > 0 i = 1 n � ( − 1 ) i a i 1 + > 0 i = 1 | a n | < 1 | J k , n − k | < | J k , 0 | , 1 ≤ k ≤ n − 2 where J k , i , k = 1 , . . . , n − 2 , i = 0 , . . . , n − k are the elements in the Jury’s array. Lagoa (Penn State) Fixed Order Interpolation Udine 2011 10 / 27

Consistency Set as a Semialgebraic Set Stability - Jury’s Criterion Hence, stability is equivalent to satisfaction of polynomial inequalities. Algebraic Description The system G ( z ) is stable if and only if p S k (Θ) > 0 , k = 1 , . . . , n + 2 . where n . p S � 1 (Θ) = 1 + a i , i = 1 n . p S � ( − 1 ) i a i , 2 (Θ) = 1 + i = 1 . p S 3 (Θ) = 1 + a n , . p S 4 (Θ) = 1 − a n , . p S J 2 k , 0 − J 2 k + 4 (Θ) = k , n − k , k = 1 , . . . , n − 2 . Lagoa (Penn State) Fixed Order Interpolation Udine 2011 11 / 27

Consistency Set as a Semialgebraic Set Dissipativity - KYP Lemma Theorem Assume that G ( z ) is stable and has no poles on the unit circle, then it is dissipative with respect to the quadratic supply function � u i � u i � T � s ( u i , y i ) . = Π , y i y i where Π ∈ R 2 × 2 is a symmetric matrix, if and only if � ∗ � 1 � 1 � Π ≤ 0 G ( z ) G ( z ) for all z ∈ C , | z | = 1 . Lagoa (Penn State) Fixed Order Interpolation Udine 2011 12 / 27

Consistency Set as a Semialgebraic Set Dissipativity - Positive Polynomials on the Unit Circle Note that � � ∗ � � 1 1 Π ≤ 0 ⇔ R ( z ) ≥ 0 G ( z ) G ( z ) where R ( z ) is the trigonometric polynomial . � � π 11 a ( z ) a ( z − 1 ) + π 12 a ( z ) b ( z − 1 ) + π 12 b ( z ) a ( z − 1 ) + π 22 b ( z ) b ( z − 1 ) R ( z ) = − Lemma A trigonometric polynomial R ( z ) is non-negative on the unit circle if and only if there exists a d r + 1 × d r + 1 positive semi-definite matrix Q . = [ q i , j ] � 0 . such that d r + k � r k = q i , i − k , k = 0 , . . . , d r . i = k Lagoa (Penn State) Fixed Order Interpolation Udine 2011 13 / 27

Consistency Set as a Semialgebraic Set Dissipativity - Positive Polynomials on the Unit Circle Note that � � ∗ � � 1 1 Π ≤ 0 ⇔ R ( z ) ≥ 0 G ( z ) G ( z ) where R ( z ) is the trigonometric polynomial d r . ✄ � � r i z i . R ( z ) = where r i = r − i ✂ ✁ k = − d r Lemma A trigonometric polynomial R ( z ) is non-negative on the unit circle if and only if there exists a d r + 1 × d r + 1 positive semi-definite matrix Q . = [ q i , j ] � 0 . such that d r + k � r k = q i , i − k , k = 0 , . . . , d r . i = k Lagoa (Penn State) Fixed Order Interpolation Udine 2011 13 / 27

Consistency Set as a Semialgebraic Set Dissipativity - Positive Polynomials on the Unit Circle Note that � � ∗ � � 1 1 Π ≤ 0 ⇔ R ( z ) ≥ 0 G ( z ) G ( z ) where R ( z ) is the trigonometric polynomial d r . ✄ � � r i z i . R ( z ) = where r i = r − i ✂ ✁ k = − d r Lemma A trigonometric polynomial R ( z ) is non-negative on the unit circle if and only if there exists a d r + 1 × d r + 1 positive semi-definite matrix Q . = [ q i , j ] � 0 . such that d r + k � r k = q i , i − k , k = 0 , . . . , d r . i = k Lagoa (Penn State) Fixed Order Interpolation Udine 2011 13 / 27