LMI based output-feedback controllers: γ-optimal versus linear quadratic.

Transcription

1 Proceedings of the 17th World Congress he International Federation of Automatic Control Seoul Korea July LMI based output-feedback controllers: γ-optimal versus linear quadratic. Dmitry V. Balandin Mark M. Kogan Nizhny Novgorod State University Nizhny Novgorod 6395 Russia tel: ; Architecture and Civil Engineering University Nizhny Novgorod 6395 Russia tel: ; Abstract: Solutiontothelinearquadraticcontrolproblem is givenintheclass oflineardynamic output-feedback full order controllers. Necessary and sufficient conditions for existence of such an optimal controller are stated in terms of linear matrix inequalities provided that initial conditions for controller states to be zero. It is shown that parameters of the optimal controller depend on an initial plant state. As an alternative we introduce γ-optimal controller which minimizes the maximal ratio of the performance index and square of the norm of the initial plant state. Numerical comparison for two kinds of these controllers is presented for inverted and double inverted pendulums. 1. INRODUCION he classical deterministic linear quadratic LQ control problem is well known see Kwakernaak and Sivan 1972 to be solvable in terms of Riccati equations only when plant state is measurable. he essential step for solving the LQ output-feedback problem was connected with presentation of the LQ performance as H 2 -norm of the system transfer matrix see Doyle et al Scherer et al he latter paper provides a result that allows to step from the performance analysis conditions formulated in terms of matrix inequalities to the corresponding linear matrix inequalities LMIs for H and H 2 synthesis problems. his is achieved by a nonlinear bijective transformation of the controller parameters. Another approach to LMI based synthesizing H -controllers utilizes so called eliminationlemma toconvertperformance analysis conditions intolmis withrespectto Lyapunovfunctionmatrix and controller parameters separately see Gahinet and Apkarian Iwasaki et al used H 2 -presentation to formulate a suboptimal LQ control problem in terms of linear matrix inequalities and synthesized a stabilizing static outputfeedbackcontrollerwhichcanguarantee aspecifiedlevelof the LQ performance for all initial conditions of the plant. Such a suboptimal controller was shown to exist if and only if there exists a positive definite matrix satisfying two LMIs while its inverse matrix satisfies another LMI. he set of these matrices satisfying the LMIs is not convex. hat is the main difficulty for solving LQ and other control problems for static or reduced order output-feedback controllers. Many computational algorithms have been developedtoovercome thisdifficulty see forexampleiwasaki and Skelton 1995 El Ghaoui et al Balandin and Kogan 24 He and Wang 26. his work was supported in part by the Russian Foundation for Basic Research Nos and he present paper deals with synthesizing full order LQ output-feedback controllers. It turns out that this problem can be formulated in terms of LMIs as a convex optimizationproblemprovidedthattheinitialstate oftheplantis known and that the initial state of the dynamic controller is zero. Necessaryandsufficientconditionsare established forexistence ofthis optimalcontroller. Itis shownthatits parameters depend on the initial state of the plant. Such a controller can be viewed as an ideal one because the initial state of the plant is not actually available for measurement. As an alternative we introduce γ-optimal output-feedback full order controller which minimizes the maximal ratio of the performance index and square of the norm of the initial plant state. Parameters of this controller do not depend on the initial state of the plant. he problem of γ-optimal control is very close to one stated by Iwasaki et al In contrast to last paper we have a convex optimization problem and can evaluate performance losses of this worst case controller comparedwiththe idealone. Some numericalexperiments with inverted and double inverted pendulums demonstrate the performance degradation of γ-optimal controller with respect to LQ ideal controller. Note that similar ideas emerged in the recent work of Köroğlu and Scherer 28 devoted to the problem of generalized asymptotic regulation with suboptimal transient response. 2. LQ OUPU-FEEDBACK CONROLLERS Consider the linear quadratic optimal control problem for the system ẋ = Ax + Bu x = x z = C 1 x + Du y = C 2 x /8/$2. 28 IFAC / KR

2 17th IFAC World Congress IFAC'8 Seoul Korea July in the class of full order dynamic output-feedback controllers ẋ r = A r x r + B r y x r = x r 2 u = C r x r + D r y providing asymptotic stability for the closed-loop system and minimizing the quadratic cost JΘ = z 2 dt 3 x R nx is the plant state u R nu is the control input y R ny is the measurable output z R nz is the controlled output x r R nx is the controller state and Ar B r Θ = C r D r are controller parameters to be found. he equation of asymptotically stable closed-loop system 1 2 is of the form ẋ c = A c x c x c = x z = C c x c x c = colx x r A + BDr C 2 BC r A c = x = B r C 2 A r C c = C 1 + DD r C 2 DC r. Since zt = C c e Act x then or JΘ = x x r 4 5 x e A c t C c C c e Act x dt 6 JΘ = x X x 7 X = e A c t C c C c e Act dt is the solution of the Lyapunov equation A c X + XA c + C c C c =. 8 hus the problem is reduced to minimization of 7 subject to constraint 8 which is a nonlinear matrix equation with respect to unknown variables X and Θ. Observe that JΘ = γ 2 x 2 and reformulate this problem as follows: given initial state of the closed-loop system x find γ = min{γ : Θ JΘ γ 2 x 2 }. 9 Matrix Θ corresponding to γ will define parameters of LQ output-feedback controller. In what follows instead of this problem we will consider the suboptimal LQ output-feedback problem: given x find γ = inf{γ > : Θ JΘ < γ 2 x 2 }. 1 Matrix Θ of the suboptimal LQ controller is defined by JΘ < γ 2 + ε x 2 for arbitrary small ε >. In fact parameters of LQ and suboptimal LQ controllers will differ indefinitely small so in the sequel we will call both controllers LQ optimal. o solve the problem 1 we will show that inequality JΘ < γ 2 x 2 11 for given γ > γ can be expressed in terms of LMIs. heorem 1. Forthesystem 4 andthecost 3 inequality 11 holds if and only if there exists matrix Y = Y > such that Ac Y + Y A c Y C c C c Y Y x < x γ 2 x 2 >. 12 Proof. Let the inequality 11 hold then x X x < γ 2 x 2 X is the solution of the matrix equation 8. Consider the equation A c X + XA c + C c C c + ε 2 I = that has an unique solution X > X and choose the parameter ε so that x X x < x X x < γ 2 x 2. hus we have A c X + XA c + C c C c < x X x < γ 2 x By multiplying the first inequality of 13 with Y = X 1 on the left and on the right we get Y A c + A c Y + Y C c C c Y < x Y 1 x < γ 2 x 2. Finally takingintoaccount Schurlemmawe arrive at 12. Now let inequalities 12 hold. heninequalities 13 with X = Y 1 hold as well and hence JΘ = x X x < x Xx < γ 2 x 2. his completes the proof. Further let us present the matrices of the closed-loop system in the form A c = A + BΘC C c = C + DΘC A nx n x nx n x B A = B = nx n x nx n x I nx C = nx n x I nx C 2 ny n x D = nz n x nx n u C = C 1 nz n x D. 14 By inserting these expressions into the first inequality of 12 we present it in the form Ψ + P Θ Q + Q ΘP < 15 Ψ = A Y + Y A Y C C Y P = CY Q = B D

3 17th IFAC World Congress IFAC'8 Seoul Korea July hen by elimination lemma see for instance Gahinet and Apkarian 1994 inequality 15 holds for some Θ if and only if A Y + Y A WP Y C W P < C Y 17 A Y + Y A WQ Y C W Q < C Y W P and W Q denote any bases of the null spaces of the matrices P and Q respectively. Observe that Y P = CY = G G = C. I Hence Y 1 W P = W G I W G denotes any basis of the null space of the matrix G. Consequently the first inequality of 12 is equivalent to following two LMIs A WG X + XA C W G < C 18 A Y + Y A WQ Y C W Q < C Y X = Y 1. oexpress 18 interms oftheplantparameters partition X and Y as X11 X 12 Y11 Y 12 X = Y =. X12 X 22 Y12 Y 22 Meanwhile since nx n x I nx nx n z G = C 2 ny n x ny n z nx n x I nx nx n z Q = B nu n x D bases of null spaces of these matrices will be of the form W C2 W 1 Q W G = W Q = I W 2 Q W 1 Q W 2 Q is any basis of the null space of B D. his reduces 18 to W G W Q A X 11 + X 11 A A X 12 C 1 W G < Y 11 A + AY 11 AY 12 Y 11 C1 Y12C 1 W Q < denotes the correspondingblockofthesymmetrical matrix. Observe that the second rows of W G and W Q are identically zero these conditions are reduced to A X M 11 + X 11 A C1 C 1 Y11 A + AY N 11 Y 11 C1 C 1 Y 11 M < N < 19 M and N are matrices whose columns form bases of the null spaces of C 2 and B D respectively. Furthermore according to Frobenius formula Y = X 1 implies Y 11 = X 11 X 12 X 1 22 X which shows there exist reciprocal matrices X > Y > with given blocks X 11 = X11 > Y 11 = Y11 > if and only if X 11 Y11 1 i.e. X11 I. 21 I Y 11 In the case of strict inequality 21 blocks Y 12 and Y 22 of the corresponding matrix Y can be chosen for example as it follows from formula X 11 = Y 11 Y 12 Y22 1 Y 12 1 in the form Y 12 = Y 22 = Y 11 X Second inequality in 12 in view of Schur lemma is equivalent to γ 2 x 2 + x r 2 x X11 X 12 x x r X12 X 22 x r >.23 hus to solve the problem 1 in accordance with heorem 1 one should find a minimal value of γ for which LMIs and 23 are feasible provided that equality 2 holds. Due to this very equality the problem under study cannot be solved generally speaking in terms of LMIs. However in the particular case when the initial state of controller is zero i.e. x r = inequality 23 is reduced to x X 11 x < γ 2 x 2 which is LMI in variables X 11 and γ 2 only and does not involve unknown matrices X 12 and X 22. Now we are in position to formulate the main result. heorem 2. LQoutput-feedbackcontrolleroftheform 2 with x r = exists if and only if LMIs A X M 11 + X 11 A C1 C 1 M < 24 Y11 A + AY N 11 Y 11 C1 N < 25 C 1 Y 11 X11 I 26 I Y 11 x X 11 x < γ 2 x 2 27 are feasible in variables X 11 = X 11 > Y 11 = Y 11 > and γ 2 >. 997

4 17th IFAC World Congress IFAC'8 Seoul Korea July Given x LQ output-feedback controller can be obtained numerically as follows: one should find the minimal γ 2 and corresponding matrices X 11 Y 11 satisfying LMIs in heorem 2 then reconstruct matrix Y by using for example 22 and finally find matrix Θ of the controller as a solution to LMI 15 under Y constructed. Note that by Finsler lemma from 24 it follows that A X 11 + X 11 A µc2 C 2 C1 < C 1 for some µ >. It means that the left upper block of the latter matrix is definitely negative i.e. the pair AC 2 should be detectable. Analogously 25 implies stabilizability of the pair AB. 3. γ-opimal OUPU-FEEDBACK CONROLLER Along with the LQ problem 1 let us consider the problem of minimizing the maximal ratio of the performance index and square of the norm of the initial plant state: find γ = inf{γ > : Θ JΘ < γ 2 x 2 x }. 28 Define γ-optimal control law of the form 2 with x r = providing JΘ < γ 2 + ε x 2 x for arbitrary indefinitely small ε >. his control law can be interpreted as minimax one since it minimizes the worst relative performance value i.e. when the initial plant state results in the maximal relative performance value. In contrast to the above case of LQ controller inequality 27 should hold now for all nonzero initial plant states which leads to inequality X 11 < γ 2 I. 29 heorem 3. γ-optimal output-feedback controller of the form 2 with x r = exists if and only if LMIs are feasible in variables X 11 = X11 > Y 11 = Y11 > and γ 2 >. Matrix of parameters Θ of γ-optimal controller is computed as provided by the procedure described above for LQ controller. Along with LQ and γ-optimal output-feedback controllers it is interesting to introduce an average LQ outputfeedback controller of the form 2 with zero initial controller state. his controller provides minimum of the expectedvalueofthe performance indexwheninitialplant state x is assumed to be a zero mean random vector satisfying E{x x } = I. In this case we have E{JΘ} < γ 2 E x 2 = γ 2 n x and consequently the average LQ output-feedback controller exists if and only if LMIs and trx 11 < γ 2 n x are feasible in variables X 11 = X11 > Y 11 = Y11 > and γ 2 >. 4. NUMERICAL COMPARISON OF LQ AVERAGE LQ AND γ-opimal OUPU-FEEDBACK CONROLLERS Let us compare LQ and γ-optimal controllers for a linear model of an inverted controlled pendulum Fig. 1. values corresponding to LQ solid line and γ-optimal dashed line controllers for the inverted pendulum without damping ẋ 1 = x 2 ẋ 2 = x 1 + u y = x 1 z 1 = x 1 z 2 = x 2 z 3 = u. he initial conditions of the plant are chosen in the form x 1 = cos ϕ x 2 = sinϕ ϕ [π]. he plot of optimal values of LQ performance as a function of angle ϕ under associated LQ controllers is shown in Fig.1 by solid line while the plot of performance values under γ-optimal controller is shown by dashed line. For example for ϕ 1 = 2π/3 and ϕ 2 = π we calculated the following parameters of LQ controllers Θ 1 = Θ 2 = hese data allow to conclude that LQ output-feedback controller considerably depends on the initial conditions of the plant. On the other hand parameters of γ-optimal controller are as follows Θ = As it follows from Fig.1 performance values corresponding to LQ and γ-optimal controllers may significantly differ. However next example of an inverted pendulum with damping ẋ 1 = x 2 ẋ 2 = x 1 x 2 + u y = x 1 z 1 = x 1 z 2 = x 2 z 3 = u shows that this difference may be not too much Fig.2. One more example is a double inverted pendulum described by equations 998

5 17th IFAC World Congress IFAC'8 Seoul Korea July CONCLUSION Fig. 2. values corresponding to LQ solid line and γ-optimal dashed line controllers for the inverted pendulum with damping Fig. 3. values corresponding to LQ solid line γ-optimal dashed line and averaged LQ dotted line controllers for the double inverted pendulum ẋ 1 = x 3 ẋ 2 = x 4 ẋ 3 = 2x 1 x 2 + u ẋ 4 = 2x 1 + 2x 2 y = x 1 z i = x i i = 14 z 5 = u. he initial conditions of the plant are chosen in the form x 1 = cos ϕ x 3 = sinϕ x 2 = x 4 = ϕ [π]. he plot of optimal values of LQ performance as a function of angle ϕ under associated LQ controllers is shown in Fig.3 by solid line while the plots of performance values underγ-optimalcontrollerandaverage LQoutputfeedbackcontrollerare shownbydashedanddottedlines respectively. In this paper the LQ output-feedback control problem is solved in the class of linear dynamic full order controllers with zero initial state. Based on LMI technique necessary and sufficient conditions for existence of LQ outputfeedback controller are derived and a computational procedure to find their parameters is given. It is shown that parameters of this controller essentially depend on the initial plant state. As an alternative γ-optimal control law is introduced which minimizes the maximal relative performance. In contrast to LQ output-feedback controller parameters of γ-optimal controller do not depend on the initial state of the plant. Numerical results show that in some cases performance losses under γ-optimal controllers may be insignificant. REFERENCES D.V. Balandin and M.M. Kogan. An optimization algorithm for checking feasibility of robust H -control problem for linear time-varing uncertain system. International Journal of Control 77: J.C. Doyle K. Glover P.P. Khargonekar and B.A. Francis. State space solutions to standard H 2 and H control problems. IEEE rans. Automat. Control 34: L. El Ghaoui F. Oustry and M.A. Rami. A cone complementarity linearization algorithm for static outputfeedback and related problems. IEEE rans. Automat. Control 42: P. Gahinet and P. Apkarian. A Linear matrix inequality approach to H control. International Journal of Robust and Nonlinear Control 4: Y. He and Q.-G. Wang. An improved ILMI method for static output feedback control with application to multivariable PID control. IEEE rans. Automat. Control 51: Iwasaki R.E. Skelton and J.C. Geromel. Linear quadratic suboptimal control with static output feedback. Systems and Control Letters 23: IwasakiandR.E. Skelton. he XY-centeringalgorithm forthe duallmiproblem: a new approach tofixed order control design. International Journal of Control 62: H. Köroğlu and C.W. Scherer. Generalized asymptotic regulation with guaranteed H 2 performance: an LMI solution. Automatica 28 submitted. H. Kwakernaak and R. Sivan. Linear Optimal Control Systems. Wiley-Interscience C. Scherer P. Gahinet and M. Chilali. Multiobjective output-feedback control via LMI optimization. IEEE rans. Automat. Control 42: