✬ ✩ Chapter 8 Reference Introduction – Integral Control Reference Input – Zero Design Motivation A controller obtained by combining a control law with an estimator is essentially a regulator design : the charac- teristic equations of the controller and the estimator are basically chosen for good disturbance rejection. However, it does not lead to tracking, which is evidenced by a good transient response of the combined system to command changes. A good tracking performance is obtained by properly introducing the reference input into the system. This is equivalent to design proper zeros from the reference input to the output. ✫ ✪ CACSD pag. 217 ESAT–SCD–SISTA
✬ ✩ Reference input – full state feedback Discrete-time : The reference signal r k is typically the signal that the out- put y k is supposed to follow. To ensure zero steady-state error to a step input r k , the feedback control law has to be modified. Modification of the control law : • Calculate the steady-state values x ss and u ss of the state x k and the output y k for the step reference r ss (=the steady-state of step reference r k ) : x ss = Ax ss + Bu ss r ss = Cx ss + Du ss Let x ss = N x r ss and u ss = N u r ss , then � � � � � � A − I B 0 N x = C D N u I ⇒ � � � � − 1 � � A − I B 0 N x = N u C D I ✫ ✪ CACSD pag. 218 ESAT–SCD–SISTA
✬ ✩ • Modify the control law: u k = N u r k − K (ˆ x k − N x r k ) = − K ˆ x k +( N u + KN x ) r k � �� � ¯ N In this way the steady-state error to a step input will be 0. Proof : 1. Verify that the closed-loop system from r k to y k is given by � � � � � � � � B ¯ − BK x k +1 A x k N = + r k , B ¯ ˆ LC A − BK − LC ˆ x k +1 x k N � � � � x k + D ¯ y k = C − DK Nr k . ˆ x k 2. If | eig( A − BK ) | < 1 and | eig( A − LC ) | < 1 we obtain the following steady-state equations : x ss + B ¯ x ss = Ax ss − BK ˆ Nr ss x ss = x ss ˆ y ss = ( C − DK ) x ss + D ¯ Nr ss u ss = − Kx ss + ¯ Nr ss ⇓ y ss = Cx ss + D ( − Kx ss + ¯ Nr ss ) = Cx ss + Du ss = r ss ✫ ✪ CACSD pag. 219 ESAT–SCD–SISTA
✬ ✩ So the transfer matrix relating y and r is a unity matrix at DC ⇒ zero steady–state tracking error, steady–state decoupling. Note that: • r k is an exogenous signal, the reference introduction will NOT affect the poles of the closed-loop system. � � − 1 A − I B • must exist, and thus for MIMO C D number of references = number of outputs • also for MIMO, reference introduction implies a steady- state decoupling between different reference and output pairs. This means that y ss = r ss . • some properties of this controller are discussed on page 227. Continuous-time : Try to verify that in this case � � � � − 1 � � 0 N x A B = N u C D I ✫ ✪ CACSD pag. 220 ESAT–SCD–SISTA
✬ ✩ There are two types of interconnections for reference input introduction with full state-feedback : Type I: u k = N u r k − K (ˆ x k − N x r k ) Type II: u k = − K ˆ x k + ( N u + KN x ) r k � �� � ¯ N y y r u r u + ¯ + N u N Plant Plant - - K K Estimator Estimator x ˆ N x x ˆ - + For a type II interconnection, the control law K used in the feedback ( u k = − K ˆ x k ) and in the reference feedforward ( ¯ N = N u + KN x ) should be exactly the same, otherwise there is a steady-state error. There is no such problem in type I. ⇒ Type I is more ROBUST to parameter errors than Type II. ✫ ✪ CACSD pag. 221 ESAT–SCD–SISTA
✬ ✩ Reference Input - General Compensator Plant and compensator model : Plant : x k +1 = Ax k + Bu k , y k = Cx k + Du k ; Compensator : x k +1 = ( A − BK − LC + LDK )ˆ ˆ x k + Ly k , u k = − K ˆ x k The structure of a general compensator with reference in- put r : u y Process r + ¯ N + M x ˆ − K Estimator ✫ ✪ CACSD pag. 222 ESAT–SCD–SISTA
✬ ✩ The general compensator is defined by the following closed- loop equations from r k to y k : � � � � � � � � B ¯ − BK x k +1 A x k N = + r k , ˆ LC A − BK − LC ˆ x k +1 x k M � � � � x k + D ¯ y k = C − DK Nr k . ˆ x k Hence, the equations defining the compensator are x k +1 = ( A − BK − LC + LDK )ˆ ˆ x k + Ly k +( M − LD ¯ N ) r k , x k + ¯ u k = − K ˆ Nr k where M ∈ R n × m and ¯ N ∈ R p × m . The estimator error dynamics are x k + B ¯ x k +1 = ( A − LC )˜ ˜ Nr k − Mr k . ✫ ✪ CACSD pag. 223 ESAT–SCD–SISTA
✬ ✩ Poles: Characteristic equation: � � �� A − BK det zI − = 0 . LC A − BK − LC This is the same characteristic equation as without ref- erence introduction. So introducing references will NOT change the poles. Zeros : The equations for a transmission zero are (see page 82) − B ¯ ζI − A BK N det = 0 − LC ζI − A + BK + LC − M D ¯ − DK C N ⇔ − B ¯ ζI − A BK N u = 0 − LC ζI − A + BK + LC − M v D ¯ C − DK N w � �� � � =0 ⇔ ✫ ✪ CACSD pag. 224 ESAT–SCD–SISTA
✬ ✩ � � � � ζI − A − B ζI − A + BK + LC − M det det = 0 ¯ − K C D N The first term determines the transmission zeros of the open loop system while the second term corresponds to the trans- mission zeros of the compensator from r k to u k : x k + ( M − LD ¯ x k +1 = ( A − BK − LC + LDK )ˆ ˆ N ) r k , x k + ¯ u k = − K ˆ Nr k These transmission zeros are designed via reference intro- duction. ✫ ✪ CACSD pag. 225 ESAT–SCD–SISTA
✬ ✩ Autonomous estimator (cfr. pag. 218-220) : Select M and ¯ N such that the state estimator error equa- tion is independent of r ⇒ M = B ¯ N where ¯ N is determined by the method for introducing the reference input with full state feedback. u y Process r + ¯ N + x ˆ − K Estimator ✫ ✪ CACSD pag. 226 ESAT–SCD–SISTA
✬ ✩ Zeros : The transmission zeros from r k to u k in this case are deter- mined by det( ζI − A + LC ) = 0 which is the characteristic equation for the estimator, hence the transmission zeros from r k to u k cancel out the poles of the state estimator. Properties : • The compensator is in the feedback path. The refer- ence signal r k goes directly into both the plant and the estimator. • Because of the pole-zero cancelation which causes “un- controllability” of the estimator modes, the poles of the transfer function from r k to y k consist only of the state feedback controller poles (the roots of det( sI − A + BK ) = 0). • The nonlinearity in the input (saturation) cancels out in the estimator since in this case the state estimator error equation is independent of u (˜ x k +1 = ( A − LC )˜ x k ) ✫ ✪ CACSD pag. 227 ESAT–SCD–SISTA
✬ ✩ Tracking–error estimator Select M and ¯ N such that only the tracking error, e k = ( r k − y k ), is used in the controller. ¯ ⇒ N = 0 , M = − L u y Process + r − − e ˆ x − K Estimator The control designer is sometimes forced to use a tracking– error estimator, for instance when the sensor measures only the output error. For example, some radar tracking sys- tems have a reading that is proportional to the pointing error, and this error signal alone must be used for feedback control. ✫ ✪ CACSD pag. 228 ESAT–SCD–SISTA
✬ ✩ Zeros : The transmission zeros from r k to y k are determined by � � � � ζI − A − B ζI − A + BK + LC L det det = 0 − K 0 C D ⇔ � � � � ζI − A − B ζI − A L det det = 0 . − K 0 C D Once K and L are fixed by the control and estimator de- sign, so are the zeros. So there is no way to choose the zeros. Properties : • The compensator is in the feedforward path. The ref- erence signal r enters the estimator directly only. The closed-loop poles corresponding to the response from r k to y k are the control poles AND the estimator poles (the roots of det( sI − A + BK ) det( sI − A + LC ) = 0). • In general for a step response there will be a steady-state error and there will exist a static coupling between the input-output pairs. • Used when only the output error e k is available. ✫ ✪ CACSD pag. 229 ESAT–SCD–SISTA
✬ ✩ Zero-assignment estimator (SISO) : Select M and ¯ N such that n of the zeros of the overall transfer function are placed at desired positions. This method provides the designer with the maximum flexibil- ity in satisfying transient-response and steady-state gain constraints. The previous two methods are special cases of this method. Zeros of the system from r k to u k : � � ζI − A + BK + LC − M det = 0 ¯ − K N ∆ ¯ = M ¯ N − 1 ⇓ M λ ( ζ ) ∆ = det( ζI − A + BK + LC − ¯ MK ) = 0 ✫ ✪ CACSD pag. 230 ESAT–SCD–SISTA
✬ ✩ Solution : ¯ Determine M using a estimator pole-placement strategy for “system” ( A z , C z ), with A z = A − BK − LC, C z = K, ¯ N is determined such that the DC gain from r k to y k is unity. For instance, in the case of a SISO system in continuous time, for which D = 0 1 ¯ N = − C ( A − BK ) − 1 B [1 − K ( A − LC ) − 1 ( B − ¯ M )] and finally M = ¯ M ¯ N . ✫ ✪ CACSD pag. 231 ESAT–SCD–SISTA
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