[k,g,gfin] = hinfsyn(p,nmeas,ncon,gmin,gmax,tol)

Transcription

1 8 H Controller Synthesis: hinfsyn The command syntax is k,g,gfin = hinfsyn(p,nmeas,ncon,gmin,gmax,tol) associated with the general feedback diagram below. e y P d u e G = F L (P, K) d K hinfsyn calculates a controller which stabilizes the system, and renders the H norm of the transfer function from d to e small. The returned variables include the closed-loop system g, controller k. The variable gfin is a scalar, containing the the H norm of the transfer function from d to e using the controller k. The program hinfsyn automatically iterates on the γ value, tightly converging to two values of γ for which a pass and fail occur. This determines γ opt to within as tight a tolerance as desired. The iteration starts from a upper and lower bound, gmax and gmin, and iterates (using bisection) until a stopping criterion is met. The stopping criterion requires the absolute difference between passing and failing values of γ to be less than tol. If the routine is called with gmax = gmin, then no iteration is performed, and only that particular value of γ is tested. The theorems derived in DGKF give necessary and sufficient conditions on the existence of a controller which renders the closed-loop d e norm less than γ. Two Hamiltonian matrices (ie., associated algebraic Riccati equations) must have positive-semi-definite stabilizing solutions, and moreover, the two solutions must satisfy a joint condition. The conditions checked for the existence of a solution are: X and Y Hamiltionian matrices must have no imaginary axis eigenvalues and the invertibility property of the stable invariant subspace must hold. The imaginary-axis eigenvalues are tested specifically min(abs(real(eig(ham)))) > epr. The conditions above imply stabilizing solutions to the Riccati equations exist. Denote these as X and Y. These must be positive-semidefinite, and they often will have eigenvalues at 0, which numerically show up as very small negatives. Hence eigenvalues of X and Y must be greater than -(epp). Finally, the coupling condition must hold, which is that the spectral radius of (X Y ) must 222

2 be less than or equal to γ 2. The selection of epr and epp to avoid false passes or fails is delicate, and at first you should just use the default values. The following assumptions are made in the implementation of the hinfsyn algorithm and must be satisfied: (i) (ii) (iii) (iv) (A, B 2 ) is stabilizable and (C 2, A) detectable (obvious) D 12 and D 21 have full rank (column, row, respectively) A jωi B2 has full column rank for all ω R. C 1 D 12 A jωi B1 has full row rank for all ω R. C 2 D 21 If the system does not satisfy these conditions, various failures occur. This is the most common problem that first-time users have. There are a few extra input and output parameters to hinfsyn, which determine some of the numerical properties and decisions that are made during calculations. k,g,gfin,ax,ay,hamx,hamy = hinfsyn(p,nmeas,ncon,gmin,gmax,tol,ricmethd,epr,epp) Input arguments p nmeas ncon gmin gmax tol open-loop, generalized plant number of measurements available to controller number of control inputs generated by controller Bisection lower bound on γ Bisection upper bound on γ Absolute stopping criteria for Bisection ricmethod 1. Eigenvalue decomposition (optional) epr epp Output arguments k g gfin 2. Schur decomposition (default) tolerance for determining if Hamiltonian matrices have imaginary-axis eigenvalues (default epr = 1e-10) positive definite tolerance for X and Y solution (default epp = 1e-6) H (sub) optimal, central controller closed-loop system with central controller implemented final passing γ value ax X Riccati solution as a VARYING matrix (IV is γ) ay Y Riccati solution as a VARYING matrix (IV is γ) hamx X Hamiltonian matrix as VARYING matrix (IV is γ) hamy Y Hamiltonian matrix as VARYING matrix (IV is γ) 223

3 The hinfsyn program displays several variables which can be checked to ensure that the above conditions are being satisfied. For each γ value being tested, the minimum magnitude, real part of the eigenvalues of the X and Y Hamiltonian matrices are displayed along with the minimum eigenvalue of X and Y which are the solutions to the X and Y Riccati equations, respectively. The maximum eigenvalue of X Y, scaled by γ 2, is also displayed. With enough knowledge about the theory, this additional information can aid you in the control design process. Specifically, at any given γ, If the X conditions are failing, then regardless of the measurements (even if they were noisefree state measurements) the disturbance-error map cannot be made less than γ because there is not the appropriate control authority. If the Y conditions are failing, then regardless of the control (even if B 2 = I you could directly affect all of the state derivatives, referred to as full control ) the disturbanceerror map cannot be made less than γ because there is too much noise/disturbances in the measurements, and the estimator cannot adequately determine approximate values of the states. If the X and Y conditions are passing, but the spectral radius condition on XY is failing, then the two individual points above are not at issue the appropriate estimation can be done if full control was available, and the appropriate control u could be calculated from the exact state measurement. In this case, the problem is the coupling between the two. There is not enough information in the measurements to correctly estimate the enough about the plant state to apply the appropriate control, given the dynamics of how the control enters (basically B 2 ). In most problems, this is the condition that is failing when γ approaches its minimum possible passing value. In addition to covering the mechanics of the command hinfsyn, this exercise will also explore some of the properties of the controller that DGKF returns. Recall that the theory goes like this: There exists a controller K that stabilizes the closed-loop and makes F L (P, K) < γ if and only if 3 conditions (which depend on γ and the plant data) hold. Note that if the conditions hold, then since F L (P, K) < γ is an open condition, there is a whole family of controllers which work. What is special about the controller that DGKF constructs? Considering all controllers which achieve the norm constraint, DGKF returns the controller which also minimizes log det γ 2 I G (jω)g(jω) dω Here, G is the closed-loop transfer function from d e. In the literature, this is called the central controller, and for a given γ, we will let K γ denote the central controller. It is useful to look at several things regarding K γ : We know the quantity F L (P, K γ ) will be less than γ. But how much less? Similar to above - what do the details of the closed-loop d e response look like across frequency. 224

4 Bandwidth of K γ as γ γ opt from above. Bandwidth of P K γ as γ γ opt from above. We will look at these in a specific example. 8.1 Example: H Controller Synthesis for HIMAT A control law will be synthesized for the HIMAT interconnection structure shown below. The controller is designed to make the H norm from d e small. e1 e 2 d1 d3 wdel d 2 d 4 e3 Σ Σ himat wp e 4 u1 y1 u 2 y 2 which is drawn in concise notation Figure 8: HIMAT Open-loop Interconnection Structure e y himat ic d u Control Design Design a H (sub)optimal control law for the open-loop generalized plant himat ic, with 2 sensor measurements, 4 error signals, 2 actuator inputs, 4 exogenous disturbances. Use default values for epr, epp and ricmethd. Initially, we will not make use of the iteration portion of the hinfsyn program. Set gmax and gmin equal, and the software simply checks the solvability conditions at that one value of γ (regardless of the tolerance, tol). 225

5 A H central control law will be designed and analyzed at different γ levels to help us gain an understanding of what the central controller is minimizing. mkhimat; minfo(himat); himatic; minfo(himat ic); nmeas = 2; % number of sensor measurements ncont = 2; % number of control inputs gmin = 10; % gamma value to be tested gmax = 10; % gamma value to be tested tol =.1; % tolerance on the gamma stopping value Try a single design at γ = 10. It should pass. Look at some of the properties. k10,g10 = hinfsyn(himat ic,nmeas,ncont,gmin,gmax,tol); rifd(spoles(starp(hinat ic,k10,2,2))) rifd(spoles(g10)) omega = logspace(-1,4,50); g10g = frsp(g10,omega); g10gs = vsvd(g10g); vplot( liv,m,g10gs) title( Singular value plot of g10 with k10 implemented ) vplot( liv,m,vsvd(frsp(k10,omega))) title( Singular value plot of k10 ) vplot( liv,lm,vsvd(frsp(mmult(himat,k10),omega))) title( Singular value plot of LoopGain10 ) What closed-loop disturbance-to-error norm is guaranteed to be achieved by this controller? What closed-loop disturbance-to-error norm is actually achieved by this controller? What does the closed-loop disturbance-to-error singular value plot look like across frequency? Is the closed-loop system stable and does it satisfy g10 gfin? Note the bandwidth of controller, and open-loop gain singular value plots σ i P (jω)k(jω). Let s rerun the above commands for gmin = gmax = 5, 3, 2 and 1.8. Name the closed-loop systems g5, g3, g2 and g18 respectively. Ask the same questions: 226

6 What closed-loop disturbance-to-error norm is guaranteed to be achieved by each of these controllers? What closed-loop disturbance-to-error norm is actually achieved by these controllers? Do the controllers synthesized at different γ levels have different -norms? What is the trend in these results? Now design an H controller using the γ iteration part of the program. The range of γ is selected to be between 1.0 and 10.0 with a stopping tolerance, tol, on the absolute difference betwen a pass and fail value of γ. Here, we set tol to 0.1. gmin = 1; gmax = 10; tol = 0.1; k,g = hinfsyn(himatic,nmeas,ncon,gmin,gmax,tol); Test bounds: < gamma <= gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f e e e e p e e e e p e e e e p e e e e p e e e e f e e e e p e e e e p e e e e f Gamma value achieved: The program outputs at each iteration the current γ value being tested, and eigenvalue information about the X and Y Hamiltionian matrices and X and Y Riccati solutions. At the end of each iteration a (p) denoting the tested γ value passed or a (f) denoting a failure is displayed. Upon finishing, hinfsyn prints out the γ value achieved. 8.2 References DGKF J. Doyle, K. Glover, P. Khargonekar, and B. Francis, State Space Solutions to H 2 and H Control Problems, IEEE Trans. Auto. Control, vol. 34, no. 8, pp , August,