AA/EE/ME 548: Problem Session Notes #5


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1 AA/EE/ME 548: Problem Session Notes #5 Review of Nyquist and Bode Plots. Nyquist Stability Criterion. LQG/LTR Method Tuesday, March 2, 203 Outline:. A review of Bode plots. 2. A review of Nyquist plots and Nyquist stability criterion. 3. LQG/LTR method  example. A Review of Bode Plots The frequency response of a linear system can be computed from its transfer function, by setting the input u(t) to: u(t) = exp{jωt} = cos(ωt)+jsin(ωt) () The resulting output can now be found as: y(t) = G(jω)exp{jωt} = M exp{j(ωt+ϕ)} = M cos(ωt+ϕ)+jm sin(ωt+φ) (2) where G(jω) denotes the transfer function, and M and ϕ are its gain and phase: M = G(jω) ϕ = arctan Im{G(jω)} Re{G(jω)} (3) The frequency response G(jω) can thus be represented by two curves: the gain curve and the phase curve. The gain curve represents G(jω) as a function of frequency ω, and the phase curve the ϕ as a function of frequency. 2 Nyquist Stability Criterion The Nyquist stability criterion allows us to determine if a system is stable or unstable. In addition, it also provides us with a measure of the degree of stability through the definition of stability margins. The given stability criterion also indicates how an unstable system should be changed to make it stable. Before stating the Nyquist stability criterion, let s first recall the definition of looptransfer function L(s). 2. The LoopTransfer Function Consider a block diagram, depicted in Figure, which represents the stochastic dynamical system controlled by a dynamical LQG controller. The looptransfer function of the given closedloop system can be found as: L(s) := F(s)G(s) (4) and it is obtained by breaking the feedback loop between the controller and the system. Thus, the loop transfer function is simply the transfer function from input u(t) to output y(t).
2 r(t) = LQG Controller F(s) u(t) System G(s) = C(sI  A)  B y(t) Fig.. Block diagram of a dynamic regulator, consisting of an optimal Kalman filter and an optimal LQR statefeedback controller. 2.2 The Nyquist Plot As stated in [2], the Nyquist plot of the looptransfer function L(s) is formed by tracing s C around the Nyquist D contour,, denotes as Γ C. This contour consists of the imaginary axis combined with an arc at infinity connecting the endpoints of the imaginary axis. The image of L(s), when s traverses Γ, gives a closed curve in the complex plane an that curve is referred to as the Nyquist plot for L(s). There is a bit of subtlety in the Nyquist plot when the looptransfer function has poles on the imaginary axis because the gain is infinite at the poles. To solve this problem, contour Γ C is modified to include small deviations that avoid any poles on the imaginary axis. The deviation consists of a small semicircles to the right of the imaginary axis pole location. Fig. 2. (a) The Nyquist D contour and (b) The Nyquist plot. The Nyquist contour encloses the right halfplane, with a small semicircle around any poles of L(s) on the imaginary axis (here illustrated at the origin) an an arc at infinity, represented by R. The Nyquist plot is the image of the looptransfer function L(s) when s traverses the Nyquist contour Γ in the clockwise direction. The solid line corresponds to ω > 0 and the dashed line to ω < 0. The plots are reproduced from [2], and the curve is generated for L(s) =.4 exp{ s} (s+) The Nyquist Stability Criterion We will state two versions of the Nyquist stability criteria. The first one represents the special case when looptransfer function L(s) is assumed to be stable. Theorem. (Simplified Nyquist criterion) Let L(s) be the looptransfer function for a negative feedback system (as shown in Figure ) and assume that L has no unstable poles, except for single poles on the imaginary axis. Then the closedloop system is stable if and only if the closed contour given by Ω = {L(jω) : < ω < } C has no net encirclements of the critical point s =. Theorem requires that looptransfer function L(s) has no unstable poles. In some cases, however, this requirement is not satisfied, and a more general result is needed. Nyquist originally considered this general case, which can be summarized as Theorem 2. 2
3 Theorem 2. (Nyquist s Stability Theorem) Consider a closedloop system with looptransfer function L(s) that has P poles in the region enclosed by the Nyquist contour. Let N be the net number of clockwise encirclements of critical point  by L(s) when s encircles the Nyquist contour Γ C in the clockwise direction. The closedloop system then has Z = N +P poles in the right halfplane. Note : Theorem 2 states that if L(s) has P unstable poles, then the Nyquist curve for L(s) should have P counterclockwise encirclements of the critical point  (so that N = P). In particular, this requires that the magnitude L(jω c ) > for some frequency ω c, corresponding to the crossing of the negative real axis. Note 2: As pointed out in [2], care has to be taken to get the right sign of the encirclements. The Nyquist contour has to be traversed clockwise, which means that frequency ω moves from to and N is positive if the Nyquist curve winds clockwise. If the Nyquist curve winds counterclockwise, then N will be negative (the desired case if P 0). 2.4 Stability Margins In real applications, it may not be enough to simply determine whether a system is stable or not. Very often we will also want to have some margins of stability, to describe how stable the system is and how robust it is to perturbations. One of the common ways to express these stability margins is the use of gain and phase margins, inspired by Nyquists stability criterion. It is known that an increase in controller gain simply expands the Nyquist plot radially. Similarly, an increase in the phase of the controller twists the Nyquist plot. It therefore becomes easy to read the amount of gain and phase that can be added without causing the system to become unstable from the Nyquist plot. Definition : The gain margin g m of a system is defined as the smallest amount that the openloop gain can be increased before the closedloop system goes unstable. For a system whose phase decreases monotonically as a function of frequency starting at 0 degrees, the gain margin can be computed based on the smallest frequency where the phase of the looptransfer function L(s) equals to 80 degrees. Let ω pc represent the frequency at which that happens. We refer to ω pc the phase crossover frequency. Using ω pc, the gain margin can be computed as: g m = L(jω pc ) (5) Definition 2: The phase margin is defined as the amount of phase lag required to reach the stability limit. Let ω gc denote the gain crossover frequency, the smallest frequency where the looptransferfunction L(s) has unit magnitude. Then for a system with monotonically decreasing gain, the phase margin is defined as: ϕ m = π +argl(jω gc ) (6) Note 3: The gain and phase margins have simple geometric interpretations on the Nyquist diagram of the looptransfer function. The gain margin is given by the inverse of the distance to the nearest point between  and 0, where the loop transfer function crosses the negative real axis. The phase margin is given by the smallest angle on the unit circle between  and the looptransfer function. When the gain or phase is monotonic, this geometric interpretation agrees with the formulas above. 2.5 Stability Margins from Bode Plot Formany systems,the gainand phaseand marginscanbe determined fromthe Bode plot ofthe looptransfer function. 3
4 To find the gain margin, we first find the phase crossover frequency ω pc where the phase is 80 degrees. The gain margin is the inverse of the gain at that frequency. To determine the phase margin, we first determine the gain crossover frequency ω gc, i.e., the frequency where the gain of the looptransfer function is. The phase margin is the phase of the looptransfer function at that frequency plus 80 degrees. Fig.3. Stability margins. The gain margin g m and phase margin ϕ m are shown on the Nyquist plot (a) and the Bode plot (b). The gain margin corresponds to the smallest increase in gain that creates an encirclement, and the phase margin is the smallest change in phase that creates an encirclement. Figures reproduced from [2]. 2.6 Example Consider the following stochastic dynamical system: [ ] ẋ = x+ 0 [ 0 ] u+ [ ] 0 w 0 y = [ 0 ] x+v (7) with noise covariance matrices Q n and R n given as: [ ] Q n = ρ, R n = (8) If our goal is to solve the LQG problem with cost matrices Q and R: [ ] Q = σ, R = (9) we could compute the optimal steadystate control gain K as: K = α [ ], where α := 2+ 4+ρ (0) and the optimal steadystate Kalman gain L as: [ L = β, where β := 2+ ] 4+σ () Please find the Nyquist plot of the given closedloop system if α = 8 and β = 27. Use Matlab function nyquist(), if necessary. What are the gain and the phase margins in this case? 4
5 Solution: To solve the given problem, we observe the gains are given as: K = 8 [ ] ] L = 27 ] (2) Using equations: α := 2+ 4+ρ β := 2+ 4+σ (3) we can find ρ and σ as: ρ = 252 σ = 62 (4) The looptransfer function of the closedloop system using the optimal steadystate LQG can now be found as: H LQG (s) = F(s)G(s) (5) where F(s) denotes the transfer function of the LQG controller and G(s) the transfer function of the openloop system: F(s) = K(sI A+BK +LC) L G(s) = C(sI A) B (6) The gain and phase margins can now be found using Matlab functions nyquist() and margin(). The complete code of this example is given below. %Example %%System dynamics A = [, ; 0 ]; B = [0; ]; 6 C = [, 0]; D = [0]; %%Cost matrices rho = 252 Q = rho [;] [,]; R = []; %%Noise covariance matrices sigma = 62; 6 Qn = sigma [;] [,]; Rn = []; %%Controller and observer gains alpha = 8; 2 beta = 27; Kc = alpha [,]; Kf = beta [;]; %%Open loop transfer function 26 [num P, den P] = ss2tf (A,B,C,D) ; %%Dynamical controller W = [Q, zeros (2, ) ; zeros (, 2), R]; 5
6 V = [Qn, zeros (2, ) ; zeros (, 2), Rn]; 3 [Af,Bf,Cf,Df] = lqg(a,b,c,d,w,v) ; %%Controller s transfer function [num F, den F] = ss2tf (Af, Bf, Cf, Df) ; G = tf (num P, den P) 36 F = tf (num F, den F) 4 %%Nyquist plot and margins figure () nyquist (F G) ; [Gm,Pm, Wcg, Wcp] = margin(f G) figure () margin(f G) ; 3 LQG/LTR Method  Matlab Software and Example 3. Stability Margins and Minimum Singular Values Let s recall the minimum singular value of I +H LQG (s) can be found as: inf ω σ(i +H LQG(jω)) := α (7) It can be shown the following inequality relations between gain margin, g m, phase margins, ϕ m, and the minimum singular value, α, are satisfied: 3.2 Matlab Software +α < g m < α ( ) ) cos α2 < ϕ m < cos ( α2 2 2 As shown in [3], Matlab functions ltru() and ltry() from the Robust Control Toolbox can be used to do LQG/LTR design. The first function (ltru()) does loop recovery at the input, with a fixed statefeedback matrix K, whereas the latter function does loop recovery at the output, with fixed Kalman gain L. The command for ltru() is given as: [af, bf, cf, df, svl] = ltru(sys, dim, Kc, Xi, Th, r, w, svk) where: [ ] A B Input sys defines the system to be controlled, sys :=, C D Input dim defines the size of matrix A, Input Kc is the fullstate feedback matrix, Input Xi is the nominal process noise covariance matrix Q n, Input Th is the measurement noise covariance matrix R n, Input r is a row vector of increasing fictitiousnoise coefficients Input w is a row vector of frequencies for Bode plots of the singular values of the loop gain Input svk is the singularvalue Bode plot of H LQR (s) and it may be omitted The function ltru() producesbodeplots ofthe singularvaluesofthe looptransferfunction H LQG (s) foreach of the values in vector r and returns a controller F(s), specified by a statespace realization (af,bf,cf,df), for the last value of r specified in the vector r. The singular values of H LQG (s) are stored in the output data svl. (8) 6
7 3.3 Example Let s consider the following helicopter model: ẋ = x u [ y = where the incremental outputs denote: y  the vertical velocity (knots/hr) y 2  the pitch altitude (radians) and the inputs are: u  the collective rotor thrust u 2  the differential collective rotor thrust Given cost matrices Q and R: and noise covariance matrices: w ] x (9) Q = C T C, R = I 2 2 (20) Q n = BB T, R n = I 2 2 (2) the minimum cost matrix S and the minimum error covariance matrix P were found as: P = S = (22) and the statespace representation of the optimal LQG controller is: ] ˆx = ˆx u [ y [ ] u opt = ˆx (23) The optimal looptransfer function H LQG (s) can now be computed as: H LQG (s) = F(s)C(sI A) B (24) and the closedloop statespace representation can be found as: [ẋ ] [ [ = x ˆx u x] 0 0 y] [ ] y = (25) ][ xˆx 7
8 The statespace representation is then used to plot the singular values of I + H LOG (s) versus frequency (Matlab function sigma()). The plot of the singular values if depicted in Figure 4, from which the minimum singular value is determined to be α = From this, we can obtain the bounds on the gain margin in each input channel using the following inequality: +α < g m < α (26) For the given system, we thus obtain: Consider now the problem of recovering gain margins: < l i < (27) < α < (28) Using inequality (26), that corresponds to α = Using Matlab function ltru(), the value of fictitious noise that corresponds to α = 0.65 was found to be r = 00. Fig.4. Singular values of I+H LQG(S) as functions of frequency. The xaxis represents frequency and yaxis singular values. Figure taken from [3]. Sources for Today s Lecture:. Karl J. Aestrom, Richard M. Murray Feedback Systems  An Introduction for Scientists and Engineers. Princeton University Press, 2009, Chapter 8, p Karl J. Aestrom, Richard M. Murray Feedback Systems  An Introduction for Scientists and Engineers. Princeton University Press, 2009, Chapter 9, p Peter Dorato, Chaouki T. Abdullah, Vito Cerone Linear Quadratic Control: An Introduction. Krieger Publishing, 2000 Chapter 7, p
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