CBE507 LECTURE IV Multivariable and Optimal Control. Professor Dae Ryook Yang

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1 CBE507 LECURE IV Multivariable and Optimal Control Professor Dae Ryook Yang Fall 03 Dept. of Chemical and Biological Engineering Korea University Korea University IV -

2 Decoupling Handling MIMO processes MIMO process can be converted into SISO process. Neglect some features to get SISO model Cannot be done always Decouple the control gain matrix K and estimator gain L. Depending on the importance, neglect some gains. Simpler Performance degradation Examples x x u K K K3 K 4 x u K K 0 0 x u K K K3 K 4 x 3 u 0 0 K3 K 4 x 3 x x 4 4 xc( k) Φcc Φcsxc( k) Γc xc( k) Φccxc( k) Φcsxs( k) Γcu( k) Lc( yc yc) uk ( ) s ( k ) sc ss s ( k) x Φ Φ x Γs xs ( k) Φscxc ( k) Φssxs ( k) Γsu( k) Ls ( ys ys ) Korea University IV -

3 Cost function ime-varying Optimal Control A discrete plant: x( k) Φx( k) Γu( k) N min J [ x ( k) Qx( k) u ( k) Qu( k)] u( k ) k 0 Q and Q are nonnegative symmetric weighting matrix Plant model works as constraints. Lagrange multiplier: (k) N min J [ x ( k) Qx( k) u ( k) Qu( k) λ ( k)( x( k) Φx( k) Γu( k))] u( k), x( k), λ( k) k 0 minimization J u ( k) Q λ ( k) Γ0 (control equations) u( k) J x( k) Φx( k) Γu( k) 0 (state equations) λ( k ) J x ( k) Q λ ( k) λ ( k) Φ0 (adjoint equations) x( k) Korea University IV -3

4 Control law: u( k) Q Γλ( k) Lagrange multiplier update: λ( k) Φ λ( k) Qx( k) λ( k) Φ λ( k) Φ Qx( k) Optimal control problem (wo-point boundary-value problem) x(0) and u(0) are known, but (0) is unknown. Since u(n) has no effect on x(n), (N+)=0. x( k) Φx( k) Γu( k) λ( k) Φ λ( k) Φ Qx( k) u( k) Q Γλ( k) Boundary Conditions λ( N) Qx( N) x(0) x 0 If N is decided, u(k) will be obtained by solving above two-point boundary-value problem. (Not easy) he obtained solution, u(k) is the optimal control policy. Korea University IV -4

5 Sweep method (by Bryson and Ho, 975) Assume (k) = S(k)x(k). Qu( k) Γ S( k) x( k) Γ S( k)( Φx( k) Γu( k)) u Q Γ S Γ Γ S Φx R Γ S Φx ( k) ( ( k ) ) ( k ) ( k) ( k ) ( k) where R Q Γ S( k ) Γ Solution of S(k) Discrete Riccati equation λ( k) Φ λ( k) Qx( k) S( k) x( k) Φ S( k) x( k) Q x( k) S x Φ S Φx ΓR Γ S Φx Q x ( k) ( k) ( k )( ( k) ( k ) ( k)) ( k) [ S( k) Φ S( k ) Φ Φ S( k ) ΓR Γ S( k ) Φ Q] x( k) 0 ( ) S k Φ [ S( k) S( k) ΓR Γ S( k)] Φ Q Single boundary condition: S(N)=Q. he recursive equation must be solved backward. Korea University IV -5

6 Optimal time-varying feedback gain, K(k) u( k) K( k) x( k) where ( ) [ k ( k) ] ( k) K Q Γ S Γ Γ S Φ he optimal gain, K(k), changes at each time but can be precomputed if N is known. It is independent of x(0). Optimal cost function value N J [ x ( k) Qx( k) u ( k) Qu( k) λ ( k) x( k) ( λ ( k) Q) x( k) u ( k) Qu( k)] k 0 N [ λ ( k) x( k) λ ( k) x( k)] k 0 λ (0) x(0) λ ( N ) x( N ) λ (0) x(0) x (0) S(0) x(0) Korea University IV -6

7 LQR Steady-State Optimal Control Linear Quadratic Regulator (LQR) Infinite time problem of regulation case LQR applies to linear systems with quadratic cost function. Algebraic Riccati Equation (ARE) S Φ S S ΓR Γ S Φ Q [ ] ARE has two solutions and the right solution should be positive definite. (J=x (0)S(0)x(0) is positive) Numerical solution should be seek except very few cases. Hamilton s equations or Euler-Lagrange equations x( k) Φx( k) Γu( k) Φx( k) ΓQ Γλ( k) λ( k) Φ λ( k) Φ Qx( k) x( k) ΦΓQ ( k) ΓΦ Q ΓQ ΓΦ x ( k) ( k) λ Φ Q Φ λ : System dynamics Hamiltonian matrix, H c Korea University IV -7

8 Hamiltonian matrix has n eigenvalues. (n stable + n unstable) Using z-transform zx( z) ΦX( z) ΓU( z) zi Φ ΓQ ( ) Γ X z U( z) zq ΓΛ( z) z ( z) 0 z ( z) ( z) z ( z) Q I Φ Λ Λ QX ΦΛ Characteristic equation zi Φ ΓQ Γ zi Φ ΓQ Γ det det Q z IΦ 0 z IΦ Q( ziφ) ΓQ Γ det( zi Φ)det(( z I Φ )[ I( z IΦ ) Q ( ziφ) ΓQ Γ ]) 0 det( zi Φ)det( z I Φ )det( I( z IΦ ) Q ( ziφ) ΓQ Γ ) 0 det(zi-)=(z) is the plant characteristics and det(z - I-)=(z - ). Called Reciprocal Root properties 0 he system dynamics using u(k)k x(k) will have n stable poles. Korea University IV -8

9 Eigenvalue Decomposition of Hamiltonian matrix Assume that the Hamiltonian matrix, H c, is diagonalizable. Eigenvectors of H c (transformation matrix): Solution * E 0 c c H W H W 0 E XI XO W * * * Λ I Λ O x x x x XI XOx * W W * * λ λ λ λ ΛI ΛOλ * N * x ( N) E 0 x (0) * N * λ ( N) 0 E λ (0) Since x * goes to zero as N, * (0) should be zero. * k * * k ( k) I ( k) I (0) (0) I ( k) * k * I I I I x X x X E x x E X x λ( k) Λ x ( k) Λ E x (0) λ( k) Λ X x( k) S x( k) u( k) K x( k) where K ( Q Γ S Γ) Γ S Φ Korea University IV -9

10 Cost Equivalent he cost will be dependent on the sampling time. If the cost equivalent is used, the dependency can be reduced. min J [ x ( k) Q x( k) u ( k) Q u( k)] min J [ x Q x u Q u] d N N t c 0 c c u( k) u( k) k 0 N ( ) k t N Q Q x Jc [ c c ] ( ) ( ) xq x uq ud k k kt k0 x u k0 Q Q u ( k) ( k) where Q Q Φ ( ) 0Q 0 Φ( ) Γ( ) d t c 0 Q Q Γ ( ) I 0 Qc 0 I Van Loan (978) Q Q Q 0 Φ Γ Φ Φ where Φ, and Φ Q Q 0 I c 0 Qc Computation of the continuous cost from discrete samples of the states and control is useful for comparing digital controllers of a system with different sample rates. Korea University IV -0

11 Optimal Estimation Least square estimation Linear static process: y=hx+v (v: measurement error) Least squares solution ( ) J ( ) ( ) J v v yhx yhx yhx ( H ) x HyHHxxˆ ( HH) Hy Difference between the estimate and the actual value xˆ x( H H) H ( Hxv) x( H H) H v If v has zero mean, the error has zero mean. (Unbiased estimate) Covariance of the estimate error P E{( xˆ x)( xˆ x) } E{( H H) H vv H( H H) } ( HH) HE{ vv} HHH ( ) If v are uncorrelated with one another, and all the element of v have the same uncertainty, { E vv } R I P ( H H) Korea University IV -

12 Weighted least squares ( ) J ( ) ( ) J v Wv y Hx W y Hx y Hx W ( H ) x HWyHWHxxˆ ( HWH) HWy Covariance of the estimate error P E xˆ x xˆ x E H WH H Wvv WH H WH {( )( ) } {( ) ( ) } ( HWH) HWE{ vv} WHHWH ( ) Best linear unbiased estimate A logical choice for W is to let it be inversely proportional to R. Need to have a priori mean square error (W=R ) xˆ ( H R H) H R y Covariance P ( H R H) Korea University IV -

13 Recursive least squares Problem (subscript o: old data, n: newly acquired data) yo Ho vo x yn Hn vn Best estimate of x: ˆx Ho Ro 0 Ho Ho Ro 0 yo ˆ x Hn 0 Rn Hn Hn 0 Rn yn Best estimate based on only old data xˆ xˆ xˆ n o [ HR H]ˆ x HR y P ( H R H ) o o o o o o o o o o o Correction using new data [ HR H] xˆ [ HR H HR H] xˆ HR y n n n o o o o n n n n n n xˆ H R H H R H H R y H xˆ [ o o o n n n] n n ( n n o) xˆ P H R ( y H xˆ ) P ( P H R H ) n n n n n o n o n n n Korea University IV -3

14 Kalman filter Plant: x( k) Φx( k) Γu( k) Γ w( k); y( k) Hx( k) v( k) Process and measurement noises: w(k) and v(k) Zero mean white noise E{ w( k)} E{ v( k)} 0 E{ w() i w ( j)} E{ v() i v ( j)} 0 (if i j) E{ w( k) w ( k)} R, E{ v( k) v ( k)} R Optimal estimation (M=P o, P(k)=P n, H=H n, R v =R n ) xˆ( k) x( k) L( k)( y( k) Hx( k)) where L( k) P( k) H ( k) R P( k) [ M H Rv H] Using matrix inversion lemma w v ( ) ( ) ( ) P k M k M k H ( HM( k) H R v ) HM( k) where M(k) is the covariance of the state estimate before measurement. v Korea University IV -4

15 Covariance update x( k) Φxˆ ( k) Γu( k) x( k) x( k) Φ( x( k) xˆ ( k)) Γ w( k) M( k) E{( x( k) x( k))( x( k) x( k)) } M( k) ΦP( k) Φ Γ RwΓ Kalman filter equations Measurement update xˆ( k) x( k) P( k) H ( k) R ( y( k) Hx( k)) ime update x( k) Φxˆ ( k) Γu( k) he initial condition for state and covariance should be known. Korea University IV -5 E{ Φ( x( k) xˆ ( k))( x( k) xˆ ( k)) Φ Γ w( k) w ( k) Γ } P( k) E{( x( k) xˆ ( k))( x( k) xˆ ( k)) }, R E{ w( k) w ( k)} P M M H HM H R v HM ( ) ( ) ( ) k k k ( ( k) ) ( k) M( k) ΦP( k) Φ Γ RwΓ v w

16 uning parameters Measurement noise covariance, R v, is based on sensor accuracy.» High R v makes the estimate to rely less on the measurements. hus, the measurement errors would not be reflected on the estimate too much.» Low R v makes the estimate to rely more on the measurements. hus, the measurement errors changes the estimate rapidly. Process noise covariance, R w, is based on process nature.» White noise assumption is a mathematical artifice for simplification.» R w is crudely accounting for unknown disturbances or model error. Noise matrices and discrete equivalents R E{ w( k) w ( k)}, R E{ v( k) v ( k)} w v E{ w( ) w ( )} R ( ), E{ v( ) v ( )} R ( ) wpsd vpsd When is very small compared to the system time constant ( c ), R R /, R R / R w wpsd v vpsd wpsd cew { ( t)}, Rvpsd cev { ( t)} Korea University IV -6

17 Linear Quadratic Gaussian (LQG) problem Estimator gain will reach steady state eventually. Substantial simplification is possible if constant gain is adopted. Assumption: noise has a Gaussian distribution Comparison with LQR: Dual of LQG M S S Γ Q Γ S Γ Γ S H ( k) ( k) ( k) [ ( k) ] ( k) S( k) Φ M( k) Φ Q Steady-state Kalman filter gain S Λ X M Λ X ( ) K Q Γ SΓ Γ SΦ L MH ( HMH R v ) where [X I ; I ] are the eigenvectors of H c associated with its stable eigenvalues. Assumption of Gaussian noise is not necessary, but with this assumption, the LQG become maximum likelihood estimate. P M M H HM H R HM ( k) ( k) ( k) ( ( k) v ) ( k) M( k) ΦP( k) Φ ΓR wγ ΦΓQ ΓΦ Q ΓQ ΓΦ Φ HRHΓΦ Γ R Γ HR HΦ v w v c He Φ Q Φ Φ ΓR wγ Φ I I I I Korea University IV -7

18 Implementation Issues Selection of weighting matrices Q and Q he states enter the cost via the important outputs N N J [ x ( k) Qx( k) u ( k) Qu( k)] J [ x ( k) H QHx( k) u ( k) Qu( k)] k0 k0 where Q and Q are diagonal matrices. he is a tuning parameter deciding the relative importance between errors and input movements. Bryson s rule y i,max is the maximum deviation of the output y i, and u i,max is the maximum value for the input u i. Q / y and Q =/ u, ii i,max, ii i,max Korea University IV -8

19 Pincer Procedure If all the poles are inside a circle of radius / (), every transient in the closed loop will decay at least as faster as / k. J [ x ( k) Q x( k) u ( k) Q u( k)] k 0 J [( x) Q( x) ( u) Q( u)] [ z Qzv ( k) Qv] k0 k0 where z(k)= k x(k), v(k)= k v(k). he state equation k k k k k k k k k k x( k) ( Φx( k) Γu( k)) z( k) Φ( x( k)) Γ( u( k)) z( k) Φz( k) Γv( k) State feedback control (LQR) Find the feedback gain for system (, ) k k v Kz u( k) K( x( k)) u( k) Kx( k) Choice of : x t x x k / k / ts ( s / ) (0)(/ ) 0.0 (0) Korea University IV -9

Here represents the impulse (or delta) function. is an diagonal matrix of intensities, and is an diagonal matrix of intensities.

Here represents the impulse (or delta) function. is an diagonal matrix of intensities, and is an diagonal matrix of intensities. 19 KALMAN FILTER 19.1 Introduction In the previous section, we derived the linear quadratic regulator as an optimal solution for the fullstate feedback control problem. The inherent assumption was that