CINVESTAV Department of Automatic Control November 3, 20 INVERSE MODEL APPROACH TO DISTURBANCE REJECTION AND DECOUPLING CONTROLLER DESIGN Leonid Lyubchyk National Technical University of Ukraine Kharkov Polytechnic Institute
2 CONTENTS Disturbance decoupling controller design problem UIO based disturbance observer design Structural synthesis Parametric synthesis Disturbance decoupling controller (DDC) design DDC existence conditions Structural singularity: DDC with fast filter Closed-loop system with DDC analysis
Methods of disturbance influence reduction Disturbance attenuation methods 3 - Methods LQ and LQG optimization, N. Wiener, R. Kalman - Methods H optimization, G. Zames, J. Doyle, B. Francis - Methods of invariant sets, F. Blanchini, A.B. Kurzhansky - Invariant ellipsoids technique, B.T. Polyak, A.S. Poznyak - liner matrix inequalities, S. Boyd Disturbance rejection methods - Feed-forward control, Jean-Victor Poncelet, 829 - Invariance theory, G.V. Shipanov, 939 - Combined control, A.G. Ivachnenko, 947 - Two channelship principle, B.N. Petrov, 953 - Disturbance observer, C. Johnson, 97 - Internal model control, B. Francis, W. Wonham, 976 - Disturbance absorption, Ya.Z. Tsypkin, 99
4 Disturbance rejection problem Plant (controlled object) x y Ax Cx, Bu y m Nw Mx Disturbance w w w c } Control law u F y ) w ( m { w Control process w y, performance index v ( y) Find control u u, so that v( y ) min, u u and closed-loop system x Ax BF ( ym ) Nw will be stable. v ( y ) 0 absolute invariance v ( y ) - invariance under the stability and robustness requirements Attainable level of disturbance rejection lim v( y ) v( y ) ( w, ), w w, { A, B, C}
Control structures for disturbance rejection 5 Feedback control Disturbance attenuation Feedback / feedforward control Disturbance compensation
Control system with disturbance observer structure 6
Output control problem 7 Consider a linear multivariable system described by the state-space model x( t ) Ax( t ) Bu( t ) Nf ( x( t ),t ), y ( t ) Cx( t ), y ( t ) Mx( t ), c m () q R - unknown distur- n m where x(t ) R - state vector, u(t ) R - control, f ( x(t ),t ) bance from certain class f ( x( t ),t ) = f, f c f x c f, 2 c r y ( t ) R, y m( t ) p R, - output controlled and measured variables respectively. We will assume that rank B m, rank C r, rank N q, rank M p. S ( ) CA B, Matrices CB S MN( 2 ) MA N are known as Markov parameters of system (). The integers, 2 are relative orders of control and disturbance transfer functions i.e. the minimal integers so that S CB( ) 0, S MN( 2 ) 0. 2
8 Output control problem Let the following assumptions take place: (a) rank B rank S ( ) r, CB (b) rank N rank S ( ) p. MN 2 (2) Without loss of generality for simplicity reason we will assume that 2 and use the notation S CB( ) S CB, S MN( ) S MN. The control problem is to find the control u( t ), depending from the measured variables, which ensure the reference signal reference model disturbances from certain class. Formally the control goal is * * y ( t ) tracking, which formed by the given y ( t ) A y ( t ) y ref ( t ) and disturbance f (( x ),t ) decoupling for all lim e (t ), t, where e c(t ) y (t ) y c(t ) - control error, constant. c * * - pre-established sufficiently small
Inverse model based disturbance observer design 9 The first step of the DDC design procedure is the state and disturbance observer n p design using UIO approach. Let z(t ) Rx(t ) R be an aggregated auxiliary variables, where R is the appropriate aggregate matrix such as Then the state vector estimation may be obtained as follows T T rank M R n. y m z M R x, M R ( P Q), ˆx(t ) Py m(t ) Qx(t ) (3) where matrices n p n n p P R, Q R are defined as MP I, RQ I, PM QR I, p n p n MQ 0, RP 0. p,n p n p,p (4)
Unknown input observer. Structural synthesis 0 The aggregated vector estimation x( t ) is given by minimal-order UIO x( t ) Fx( t ) G y ( t ) Hy ( t ) G u( t ). (5) m m 0 The UIO parameters are determined from invariance conditions R HM A F R HM GM, RN HMN 0,G RB 0,G G FH. 0 (6) If assumption (2b) takes place, a solution of (5) may be obtained as F RΠ AQ, G RB, G RΠ AP, N 0 H RNS, Π I BS M, MN N n MN N (7)
Inverse model-based disturbance observer Taking the unknown disturbance estimation in the form ˆ ˆ ˆ f ( t ) N x( t ) Ax( t ) Bu( t ). (8) The minimal-order state and disturbance observer (SDO) equation: x(t ) RΠN AQx(t ) RΠN APy m(t ) RNSMN y m(t ) RΠN Bu(t ), ˆf ( t ) C ( y ( t ) MAQx( t ) MAPy ( t ) S u( t )), N m m MB C S N PΩ. N MN N The estimation errors x e (t ) x(t ) x(t ˆ ), e ˆ f ( t ) f ( x,t ) f ( t ) (9) e ( t ) F R e ( t ), e ( t ) Qe ( t ), x x x x e ( t ) C MAQe ( t ). f N x (0)
Unknown input observer. Parametric synthesis Concretely define the matrices P I, Q, than with p 0 p,n p P Q P Q, P Q 2 2 R Q2 P2 I n p and P, Q 2 2 are arbitrary matrices with det Q 2 0. For system representation A A N A, M I, N A A N 2 p 0n p,p 2 22 2 the observer dynamics matrix has the form: 2 22 2 2 2 F R Q A P A Q, A A, A A N N A 2 N 2 22 22 2 2 Ω I N N. N q p n p () (2)
Unknown input observer. Parametric synthesis 3 Thus the matrix Q 2 defines the similarity transformation and doesn t change the n p p spectrum of F R, which completely determined by arbitrary matrixp 2 R. The last may be choused by pole placement method if pair ( A 22,A 2) is observable. Such a condition is equivalent to the well-known UIO design solvability condition, namely observable of matrix pair ( Π, M ). The aggregate matrix R is determined up to an N arbitrary nonsingular matrix Q 2. F R) Q( A P A ) Q ( 22 2 2 P2 Tuning parameters UIO pole placement Solvability conditions ( F( P * * * 2)) (... n p ) ( A 22, A 2 ) is observable
Regularized disturbance observer design 4 The observability condition is violated in the case when p q. Ω N 0 and F( R ) doesn t depend from P 2. At that In such singular case for the tuning properties guarantee it is possible to use the socalled regularized UIO, which ensure the appriximately invariance with respect the the unknown disturbance where 2 2 RN HCN H min (3) H 0 -regularization parameter. Then T T q MN MN N n H RNS I S S, Π I H M (4)
Regularized SDO design problem solution 5 F A P Ω A, A A N Ψ A, N 22 2 N 2 22 22 2 N 2 T T q Ψ N I N N, T T T N q q q Ω I N N I N N I N N. (5) Estimation error equations for the regularized state and disturbance observer are the following: T x x q MN MN e ( t ) F e ( t ) RN I S S f ( x,t ), e ( t ) N PΩ H MAQe ( t ) f N N x T n q MN MN N I PM I S S f ( x,t ) and for small value of may be done sufficiently small. (6)
Disturbance compensator design 6 The disturbance compensative control is a function of reference signal and disturbance estimation in the form of TDF controller. In the usual case of square plant (r m ) under the assumption (2a) * u ( t ) S ( y ( t ) C x( t ) S ˆf ( t )), C A C CA. (7) CB ref Aˆ CN A If system structure non-singularity condition take place rank S m q, S m CB CN I S S S S I MN MB q (8) then disturbance estimation may be eliminated from the controller equation and DDC has the form of two-degree-of-freedom controller.
Disturbance decoupling controller design DDC equations are: S CN 0 0 N N N m N B ref CB ref A CB A N N m x( t ) F x( t ) RΠ A ( PΩ H )y ( t ) Π H y ( t ), (9) * det S ( C, B, N, M ) 0, u ( t ) S ( y ( t ) C Qx( t )) S C ( PΩ H )y ( t )), 0 0 0 N B A B CB N MN F RΠ A Q, A A H C, H BS, H NS. det S ( C, B, N, M ) 0 The realizable controller may be obtained using the disturbance estimations dynamically transformed by the internal dynamic filter: * u ( t ) S CB( y ( t ) CAx( ˆ t ) SCN f ( t )) 0, 0 - small filter parameters (20) f ( t ) f ( t ) ( ) ˆf ( t ), DDC equations are: * u( t ) u( t ) ( )( ( t ) SCBS CN 2( t )), u ( t ) u( t ) ( t ), CB ref Aˆ 2 N m m ( t ) S ( y ( t ) C x( t )), ( t ) C ( y ( t ) MAQx( t ) MAPy ( t )). 0 7 (2)
Disturbance decoupling controller with fast filter structure 8
Closed-loop system with disturbance decoupling controller analysis 9 If system structural matrix S is nonsingular, the closed-loop system equation is: 0 0 x( t ) A x( t ) Π Nf ( t ) H y ( t ) Le ( t ), B B ref x A A H C Π A H A C, B A B B * (22) * The control goal is achieved with 0, if closed-loop system (22) is stable, e (t ) tends to zero due to properties of UIO. because x 0 For nonminimum-phase systems, matrix A is unstable. The problem of closedloop system stabilizing arises, moreover simple additional state feedback * ˆ u(t ) u (t ) Kx(t ) doesn t change closed-loop matrix spectrum because Π ( A BK ) 0. B In such a case the local optimal control method may be applied.
Local optimal control for disturbance rejection 20 2 2 y ( t ) C Ax( ˆ t ) S u( t ) S ˆf ( t ) u( t ) min (23) ref A CB CN The corresponding control law is given by u * u ( t ) D y ( t ) C Ax( ˆ t ) S ˆf ( t ) ref A CN * T T CB m CB CB CB D S u ( t ), D I S S S, (24) From (23) the equation of closed-loop system follows 0 0 x( t ) A ( )x( t ) BD y ( t ) Π Nf ( x,t ) L e ( t ), B n A B ref B x A ( ) A BD C Π A BD A C, Π I BD C. * (25)
Closed-loop system properties 2 Using the combined control * ˆ u(t ) u (t ) Kx(t ) find that T 0 0 m CB CB A (,K ) A ( ) B K, B B I S S, Closed-loop system with combined control may be stabilized, if matrix pair A 0( ), B is controllable. The control error is given by * T * c c CB m CB CB e ( t ) A e ( t ) S I S S u ( t ) (26) and control goal is achieved with * ( ). Attainable accuracy of control
22 Closed-loop two-time-scale system For the structural singular plant closed-loop system with DDC includes internal filter Singular perturbation 0 x( t ) A x( t ) Nf ( x( t ),t ) H S f ( t ) H y ( t ) Le ( t ), B CN B ref x f ( t ) f ( t ) ( ) f ( x( t ),t ) ( )e ( t ) (27) f The closed-loop system (27) is two-time scale system, in which slow motion under 0 coincides with the process in the system with ideal DDC and the fast one satisfied the dynamic equation: 0 0 E( )x( t ) A x( t ) B f ( x( t )). (28) 0 0 B CN 0 In 0 A H S N E( ), A, B. 0 I 0 I ( )I m q,n q q
Robust decoupling controller design 23 Fast motion stability problem reduced to the absolute stability problem of system (28) with nonlinearities from certain class. For the particular case of linear state-dependent uncertain disturbance f ( x(t ),t ) Ax(t ), where A, A c A is the system () dynamic matrix perturbation 0 0 A N A HBSCN A ( A ) (29) ( ) A Iq and fast motion stability analysis reduced to the robust stability problem 0 Re ( A ( )) -, c. (30) A A A
Disturbance decoupling controller existence conditions 24 Invertability conditions (a) rank B rank S ( ) r, CB (b) rank N rank S ( ) p. MN 2 Structural nonsingularity conditions rank S m q, S m C S CB CN I S S I N MB q Input (strong) observability conditions det 0, Iq CN SMBSCBSCN NA, M is observable (detectable)
DD existence conditions extension 25
Example. Magnetic suspension disturbance rejection control Linearized mathematical model of the system x ( t ) 0 0 0 x ( t ) 0 0 x 2( t ) 0 0 h x 2( t ) 0 0 u( t ) f 2( t ), x ( t ) a a a 0 x ( t ) b 3 0 2 3 f ( t ) 0 0 0 v f ( t ) 0 0 2 c m m 3 y ( t ) x ( t ), y ( t ) x ( t ), y ( t ) x ( t ) where input f ( t) ( t) and state-dependent disturbances f 2 ( t) f ( x( t), u( t)), characterized the external forces and system's non-stationary parameters variations. 26 Control problem: using the measurements y m ( t) x( t), y m ( t) x3( t) find the control function u (t) so that the controlled output y c(t ) x (t ) (deviation from the desired position) tracks the signal, generated by reference model y (t ) 2y (t ) y (t ) 0y (t ) 0. 2
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35 The designed DD controller has the structure of multivariable PI-controller with small parameters
PI controller Simulation results 36
Disturbance estimation by PI and UI observers 37
UIO based disturbance compensative controller 38
Disturbance decoupling controller 39
Example. Chaotic oscillator synchronisation 40
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Conclusion 44 Disturbance decoupling compensator (DDC) design method for multivariable systems measurements is proposed using the UIO technique. The design procedure includes state and disturbance observer design and disturbance compensator design. If system structure non-singularity conditions take place, the disturbance estimation may be eliminated from the control law and DDC equations are obtained in the explicit form. For the case when such a conditions are violated the realizable form of the DDC should be included additional internal dynamic filter with small time constant. For two-time-scale closed-loop system if the fast motion is stable the slow one coincides with the processes in the system with ideal compensator.
Thank you for your attention! 45