Chapter 6 State-Space Design

Transcription

1 Chapter 6 State-Space Design wo steps. Assumption is made that we have all the states at our disposal for feedback purposes (in practice, we would not measure all these states). his allows us to implement a control law.. Estimator or observer design he dynamic system obtained from the continued control law and estimator is called the controller Control law Consider a linear combination of the states u Kx k k his control law assumes that r and usually referred to as a regulator. We have the difference equation Substituting u- K x aking z transform he characteristic equation is x(k +)φx(k)+γu(k) x(k +)φx(k) ΓKx(k) (zi φ +ΓK)X(z) det(zi φ +ΓK) x x. Pole Placement Given desired root locations z i β, β,β 3, the desired characteristic equation is α c (z) (z β )(z β )(z β 3 )

2 Example Design a control law for the plant ẋ x ẋ x }{{} F he discrete model of the system is + u }{{} G φ e F, Γ e Fη dη G φ, Γ Suppose we wish to pick z-plane roots of the closed-loop characteristic equation so that equivalent s-plane roots have a damping ratio of ζ.5 and a real part of s.8rad/sec (i.e. s.8 ± j3.) Using z e s with. we find z.8 ± j.5 the desired characteristic equation is z.6z +.7 () ( ) det z + k k or ( ( ) ) ( ) z + k + k z + k k + () equating coefficients of like power in () and () ( ) k + k.6 ( ) k k +.7 k., k , for. he above calculations are easier if use is made of the control canonical form a a a 3 Φ c, Γ c, H c b b b 3

3 he characteristic equation of Φ c is a(z) z 3 + a z + a z + a 3 i.e. the coefficients of the characteristic polynomial are the negative of the coefficients of the first row of Φ c. he closed-loop system matrix Φ c Γ c K is Φ c Γ c K he characteristic equation is a k a k a 3 k 3 z 3 +(a + k )z +(a + k )z +(a 3 + k 3 ) If the desired root locations are given by the roots of the equation then the gains are z 3 + α z + α z + α 3 k α a ; k α a ; k 3 α 3 a 3 Steps in canonical-form design method Given an arbitrary (Φ, Γ) and desired characteristic equation α(z), by redefining the states we convert (Φ, Γ) to control form (Φ c, Γ c ) and find the gain as shown above. hen we transform the gains back in terms of the original states. Controllability Is it always possible to find an equivalent (Φ c, Γ c ) for arbitrary (Φ, Γ)? If the roots of det(zi φ) are distinct, then the state equation may be transformed into the following form x(k +) λ λ... λ n Γ x(k)+ Γ. Γ n u(k) Criterion for controllability: no element of Γ can be zero. 3

4 Ackerman s Formula Given that (Φ c, Γ c ) exists, i.e.the system is controllable, then the gains to implement a control law are given by K Γ ΦΓ Φ Γ Φ n Γ αc (Φ) where C Γ ΦΓ is the controllability matrix, n is the order of the system and we substitute Φ for z in α c (z) toform α c (Φ) Φ n + α Φ n + α Φ n + + α n I where α i s are the coefficients of the desired characteristic equation. α i (z) zi φ +Γk z n + α z n + + α n Example Redo last example α.6, α.7 Since desired characteristic equation is z.6z +.7 α c (Φ) also and Γ ΦΓ Γ ΦΓ K k k k k Which is the same result as before 4

5 Estimator Design here are two kinds of basic estimates of the state x(k):. he current estimate, ˆx(k) is based on measurements y(k) up to and including the kth instant.. predictor estimate, x(k) is based on measurements up to y(k ) We will like to set u K ˆx or u K x Prediction estimator o estimate the state we could construct a model of the plant dynamics x(k +)Φ x(k)+γu(k) We know Φ, Γandu(k), so the above system should work if we can obtain x() and set x() equal to it. Define an error in the estimate x x x ( ) 5

6 he dynamics of the estimator-error is given by x(k +)Φ x(k) if the initial value of x is off, the dynamics are those of the uncompensated plant, Φ. he estimator is running open-loop and not utilizing any continuing measurements of the system s behavior, and so we expect it to diverge from the truth. However, we can incorporate feedback to better the estimates. he closed-loop estimator is described by x(k +)Φ x(k)+γu(k)+l p y(k) H x(k) ( ) where L p is the feedback gain matrix his is a Prediction estimator because a measurement at time k results in an estimate of the state that is valid at 6

7 time k +, i.e. the estimate has been predicted one cycle in the future. he dynamics of the estimator error is found by subtracting x(k +) Φx(k)+Γu(k) y(k) Hx(k) from( ) x(k +) Φ L p H x(k) Due to feedback action, x(k) will converge to x(k) regardless of x() and will do so quickly particularly if L p is large (assuming a stable system matrix). o find L p, we do the same as for designing the control law. Specify the desired estimator root locations in the z-plane char. eqn. (z β )(z β ) (z β n ) ( ) equate the coefficients of ( ) and zi Φ+L p H Example Construct estimator for previous examples. he measurement is of x H. he desired roots are z.4 ± j.4 s-plane roots with ζ.6 and ω n that is 3 times faster than the control roots selected. Desired characteristic equation gives (approx.) z.8z +.3 or ( det z zi Φ+L p H + Lp L p ) z +(L p )z + L p + L p L P. L P.5 5. he estimator algorithm to be coded in the computer is x (k +) x (k)+.5u(k)+. x (k)+.y(k) x (k) x (k +) x (k)+.u(k)+5.y(k) x (k) 7

8 Since Φ Γ. Φ. Γ.5. Observability Given a desired set of estimator roots, is L p uniquely determined? It is provided y is a scalar and the system is observable. Ackerman s Formula L p may be determined from L p α e (Φ) H HΦ HΦ. Hφ n }{{} O. where α e (Φ) Φ n + α Φ n + α Φ n + + α n I and the α is are the coefficients of the desired characteristic equation, i.e. α e (z) z n + α z n + + α n O is the observability matrix and it must be full rank for the matrix to be invertible and for the system to be observable. 8

9 Current Estimator For the prediction estimator, the state estimate x(k) is arrived at after receiving measurements up through y(k ). he current estimator, estimates the state ˆx(k) based on the current measurement y(k). ˆx(k) x(k)+l c (y(k) H x(k)) (3) where x(k) is the predicted estimate based on a model prediction from the previous time estimate, that is (3) and (4) x(k) Φˆx(k ) + Γu(k ) (4) x(k +)Φ x(k)+γu(k)+φl c y(k) H x(k) (5) he estimation-error equation for x(k) is x(k +)Φ ΦL c H x(k) (6) where x x x Fromtheaboveweseethat x in the current estimator equation is the same quantity as x in the predictor estimator equation and the estimator gain matrices are related by L p ΦL c (7) the estimator-error equation for ˆx is Where x ˆx x x(k +)Φ L c HΦ x(k) (8) (6) and (8) can be shown to have the same roots. herefore one could use either form as the basis for computing the estimator gain, L c. Using (8) which is similar to prediction estimator result except that HΦ appears instead of H, Ackerman s formula for L c is L c α e (Φ) HΦ HΦ HΦ 3. Hφ n where α e (Φ) is based on the desired root locations. Note (7) L c Φ L p (). (9) 9

10 Example Repeat previous example using the current estimator formulation (8) L c he estimator implementation using (3) and (4) in a way that reduces the computation delay as much as possible is, before sampling and after sampling y(k) x (k) ˆx (k ) +.5u(k ) +.ˆx (k ) x (k) ˆx (k ) +.u(k ) x (.68) x (k) x x (k) 5. x (k) ˆx (k) x +.68y(k) ˆx (k) x +5.y(k) Reduced-Order Estimator Not enough time to cover it.

11 Regulator Design: Combined Control Law and Estimator he Separation Principle Setting u K x x(k +) Φx(k) ΓK x(k) Φx(k) ΓK(x(k)+ x(k)) Combining this with the estimator-error equation gives x(k +) x(k +) he characteristic equation is Φ Lp H ΓK Φ ΓK zi Φ+L p H ΓK zi Φ+ΓK x(k) x(k) zi Φ+L p H zi Φ+ΓK α e (z)α c (z) i.e. the characteristic roots of the complete system consist of the combination of the estimator roots and the control roots that are unchanged from those obtained assuming actual state feedback.

12 It is interesting to compare the results of the state-space designed compensator to that of classical compensation. Prediction estimator x(k) (Φ ΓK L p H) x(k ) + L p y(k ) Current estimator u(k) K x(k) ˆx(k) (Φ ΓK L c HΦ+L c HΓK)ˆx(k ) + L c y(k) u(k) K ˆx(k) he poles of the controllers above are obtained from or zi Φ+ΓK + L p H zi Φ+ΓK + L c HΦ L c H ΓK and are neither the control law poles nor the estimator poles. Converting the difference equations to transfer functions results in Prediction estimator U(z) Y (z) D p(z) KzI Φ+ΓK + L p H L p Current estimator U(z) Y (z) D c(z) KzI Φ+ΓK + L c HΦ L c HΓK L c z