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h2syn

H2 control synthesis for LTI plant

Syntax

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

Description

[K,CL,GAM,INFO] = H2SYN(P,NMEAS,NCON) computes a stabilizing H2 optimal controller K for a partitioned LTI plant P:

H2 control system CL = lft(P,K):

The LTI system P is partitioned such that 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). The controller, K, is a state-space (ss) model and has the same number of states as P.

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 γ in GAM. INFO is a struct array that returns additional information about the design.

Output Arguments

Description

K

LTI controller

CL= lft(P,K)

LTI closed-loop system Ty1u1

GAM = norm(CL)

H2 optimal cost γ = Ty1u12

INFO

Additional output information

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

INFO.NORMS

Norms of four 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)

u2(t)=KFIx(t)

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

collapse all

Stabilize a 4-by-5 unstable plant with three states, two measurement signals, and one control signal.

In practice, P is an augmented plant that you have constructed by combining a model of the system to control with appropriate H2 weighting functions. For this example, use a randomly-generated model.

rng(0,'twister');
P = rss(3,4,5)';

This command creates a 4-output, 5-input stable model and then takes its Hermitian conjugate. This operation yields a 5-output, 4-input unstable model. For this example, assume that one of the inputs is a control signal and two of the outputs are measurements.

Confirm that P is unstable. All the poles are in the right half-plane.

pole(P)
ans = 

    0.2593
   15.9497
   20.7994

Design the stabilizing controller, assuming NMEAS = 2 and NCON = 1.

[K,CL,GAM] = h2syn(P,2,1);

Examine the closed-loop system to confirm that the plant is stabilized.

pole(CL)
ans = 

  -26.8951
  -22.4817
  -20.6965
  -17.6041
   -0.8694
   -2.6697

Shape the singular value plots of the sensitivity and complementary sensitivity .

To do so, find a stabilizing controller K that minimizes the norm of:

Assume the following plant and weights:

Using those values, construct the augmented plant P, as illustrated in the mixsyn reference page.

s = zpk('s');
G = 10*(s-1)/(s+1)^2;
G.u = 'u2';
G.y = 'y';

W1 = 0.1*(s+1000)/(100*s+1); 
W1.u = 'y2';
W1.y = 'y11';

W2 = tf(0.1); 
W2.u = 'u2';
W2.y = 'y12';

S = sumblk('y2 = u1 - y');
 
P = connect(G,S,W1,W2,{'u1','u2'},{'y11','y12','y2'});

Use h2syn to generate the controller. Note that this system has NMEAS = 1 and NCON = 1.

[K,CL,GAM] = h2syn(P,1,1);

Examine the resulting loop shape.

L = G*K; 
S = inv(1+L); 
T = 1-S;
sigmaplot(L,'k-.',S,'r',T,'g')
legend('open-loop','sensitivity','closed-loop')

Algorithms

h2syn computes the H2 optimal controller. The standard implicit assumptions for the existence of an optimal solution are:

  • (A, B2, C2) are stabilizable and detectable.

  • P12 and P21 have no zeros on the imaginary axis (continuous time) or unit circle (discrete time).

  • D12 has full column rank and D21 has full row rank. In other words, P12 and P21 have no zeros at infinity.

If the second or third of these conditions are violated, then h2syn makes slight adjustments to the plant data to enforce these conditions. This process is called regularization. It allows h2syn to compute the optimal controller for a nearby problem even when the implicit assumptions of H2 synthesis are not all satisfied.

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

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Introduced before R2006a

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