Robust Control Toolbox™

H₂ control synthesis for LTI plant

Syntax

• [K,CL,GAM,INFO]=H2SYN(P,NMEAS,NCON)
  

Description

h2syn computes a stabilizing H₂ optimal lti/ss controller K for a partitioned LTI plant P. The controller, K, stabilizes the plant P and has the same number

of states as P. The LTI system P is partitioned where inputs to B1 are the disturbances, inputs to B2 are the control inputs, output of C1 are the errors to be kept small, and outputs of C2 are the output measurements provided to the controller. B2 has column size (NCON) and C2 has row size (NMEAS).

If P is constructed with mktito, you can omit NMEAS and NCON from the arguments.

The closed-loop system is returned in CL and the achieved H2 cost in GAM. --see Figure 2. INFO is a struct array that returns additional information about the design.

Figure 5-5: H2 control system CL= lft(P,K)=.

.

Output Arguments:
K	LTI controller
CL= lft(P,K)	LTI closed-loop system
GAM = norm(CL)	the H2 optimal cost =
INFO	additional output information

Additional output -- structure array INFO containing possible additional information depending on METHOD)

INFO.NORMS	Norms of 4 different quantities, full information control cost (FI), output estimation cost (OEF), direct feedback cost (DFL) and full control cost (FC). NORMS = [FI OEF DFL FC];
INFO.KFI	Full-information gain matrix (constant feedback)
INFO.GFI	Full-information closed-loop system GFI=ss(A-B2*KFI,B1,C1-D12*KFI,D11)
INFO.HAMX	X Hamiltonian matrix (state-feedback)
INFO.HAMY	Y Hamiltonian matrix (Kalman filter)

:

Examples

Example 1: Stabilize 4-by-5 unstable plant with 3-states, NMEAS=2, NCON=2.

• rand('seed',0);randn('seed',0); 
  P=rss(3,4,5)';
  [K,CL,GAM]=h2syn(P,2,1);
  open_loop_poles=pole(P)
  closed_loop_poles=pole(CL)
  

Example 2: Mixed-Sensitivity H₂ loop-shaping. Here the goal is to shape the sigma plots of sensitivity S:=(I+GK)^-1 and complementary sensitivity T:=GK(I+GK)^-1, by choosing a stabilizing K the minimizes the H₂ norm of

where , , no W₃.

• s=zpk('s');
  G=10*(s-1)/(s+1)^2;
  W1=0.1*(s+100)/(100*s+1); W2=0.1; W3=[];
  P=(G,W1,W2,W3);
  [K,CL,GAM]=h2syn(P);
  L=G*K; S=inv(1+L); T=1-S;
  sigma(L,'k-.',S,'r',T,'g')
  

Algorithm

The H₂ optimal control theory has its roots in the frequency domain interpretation the cost function associated with time-domain state-space LQG control theory [1]. The equations and corresponding nomenclature used here are taken from the Doyle et al., 1989 [2]-[3].

h2syn solves the H₂ optimal control problem by observing that it is equivalent to a conventional Linear-Quadratic Gaussian (LQG) optimal control problem. For simplicity, we shall describe the details of algorithm only for the continuous-time case, in which case the cost function J_LQG satisfies

.

with plant noise u₁ channel of intensity I, passing through the matrix [B1;0;D12] to produce equivalent white correlated with plant and white measurement noise having joint correlation function

The H₂ optimal controller K(s) is thus realizable in the usual LQG manner as a full-state feedback K_FI and a Kalman filter with residual gain matrix K_FC.

1. Kalman Filter

where Y= YT0 solves the Kalman filter Riccati equation

2. Full-State Feedback

where X = XT0 solves the state-feedback Riccati equation

The final positive-feedback H2 optimal controller has a familiar closed-form

h2syn implements the continuos optimal H2 control design computations using the formulae described in the Doyle, et al.[2]; for discrete-time plants, h2syn uses the same controller formula, except that the corresponding discrete time Riccati solutions (dare) are substituted for X and Y. A Hamiltonian is formed and solved via a Riccati equation. In the continuous-time case, the optimal H2-norm is infinite when the plant D₁₁ matrix associated with the input disturbances and output errors is non-zero; in this case, the optimal H2 controller returned by h2syn is computed by first setting D₁₁ to zero.

3. Optimal Cost GAM.

The full information (FI) cost is given by the equation. The output estimation cost (OEF) is given by, where . The disturbance feedforward cost (DFL) is , where L2 is defined by and the full control cost (FC) is given by . X2 and Y2 are the solutions to the X and Y Riccati equations, respectively. For for continuous-time plants with zero feedthrough term (D11 = 0), and for all discrete-time plants, the optimal H₂ cost = is

• GAM =sqrt(FI^2 + OEF^2+ trace(D11*D11'));
  

otherwise, GAM = Inf.

Limitations

• (A, B₂, C₂) must be stabilizable and detectable.
• D₁₂ must have full column rank and D₂₁ must have full row rank

References

[1]  Safonov, M.G., A.J. Laub, and G. Hartmann, "Feedback Properties of Multivariable Systems: The Role and Use of Return Difference Matrix," IEEE Trans. of Automat. Contr., AC-26, pp. 47-65, 1981.

[2]  Doyle, J.C., K. Glover, P. Khargonekar, and B. Francis, "State-space solutions to standard H2 and H control problems," IEEE Transactions on Automatic Control, vol. 34, no. 8, pp. 831-847, August 1989.

[3]  Glover, K., and J.C. Doyle, "State-space formulae for all stabilizing controllers that satisfy an H norm bound and relations to risk sensitivity," Systems and Control Letters, 1988. vol. 11, pp. 167-172, August 1989.

See Also
augw        Augment plant weights for control design

hinfsyn     H synthesis controller

Provide feedback about this page

h2hinfsyn

hankelmr

© 1984-2008- The MathWorks, Inc.    -   Site Help   -   Patents   -   Trademarks   -   Privacy Policy   -   Preventing Piracy   -   RSS