Topic #5 6.3 Feedback Control State-Space Systems Full-state Feedback Control How do we change the poles of the state-space system? Or,evenifwecanchangethepolelocations. Where do we put the poles? Linear Quadratic Regulator Symmetric Root Locus How well does this approach work? Copyright 2 by Jonathan How. All Rights reserved
Fall 2 6.3 5 Pole Placement SofarwehavelookedathowtopickK to get the dynamics to have some nice properties (i.e. stabilize A) λ i (A) ; λ i (A BK) Classic Question: whereshouldweputtheseclosed-looppoles? Of course we can use the time-domain specifications to locate the dominantpoles rootsof: s 2 +2ζω n s + ω 2 n = Then place rest of the poles so they are much faster than the dominant behavior. For example: Could keep the same damped frequency w d andthenmovethe real part to be 2 3 times faster than real part of dominant poles ζω n Just be careful moving the poles too far to the left because it takes a lot of control effort
Fall 2 6.3 5 2 Could also choose the closed-loop poles to mimic a system that has similar performance to what you would like to achieve: Just set pole locations equal to those of the prototype system. Various options exist Bessel Polynomial Systems of order k G p (s) = B k (s) All scaled to give settling times of second, which you can change to t s by dividing the poles by t s.
Fall 2 6.3 5 3 Procedure for an n th order system: Determine the desired settling time t s Find the k = n polynomial from the table. Divide pole locations by t s Form desired characteristic polynomial Φ d (s) and use acker/place to determine the feedback gains. Simulate to check performance and control effort. Example: with so that n = k =3. A = G(s) = 5 4 s(s +4)(s + ) B = Want t s = 2 sec. So there are 3 poles at: 5.93/2 = 2.547 and ( 3.9668 ± 3.7845i)/2 =.9834 ±.8922i Use these to form Φ d (s) andfind the gains using acker The Bessel approach is fine, but the step response is a bit slow.
Fall 2 6.3 5 4 Another approach is to select the poles to match the n th polynomial that was designed to minimize the ITAE integral of the time multiplied by the absolute value of the error J ITAE = in response to a step function. Z t e(t) dt Both Bessel and ITAE are tabulated in FPE-58. Comparison for k =3(Givenforω = rad/sec, so slightly different than numbers given on previous page) φ B d =(s +.942)(s +.7465 ±.72i) φ ITAE d =(s +.78)(s +.52 ±.68i) So the ITAE poles are not as heavily damped. Some overshoot Faster rise-times. Problem with both of these approaches is that they completely ignore the control effort required the designer must iterate.
Fall 2 6.3 5 5 Linear Quadratic Regulator An alternative approach is to place the pole locations so that the closed-loop (SISO) system optimizes the cost function: J LQR = Where: Z x T (t)(c T C)x(t)+ru(t) 2 dt y T y = x T (C T C)x {assuming D =} is called the State Cost u 2 is called the Control Cost, and r is the Control Penalty Simple form of the Linear Quadratic Regulator Problem. Can show that the optimal control is a linear state feedback: u(t) = K lqr x(t) K lqr found by solving an Algebraic Riccati Equation (ARE). We will look at the details of this solution procedure later. For now, let s just look at the optimal closed-loop pole locations.
Fall 2 6.3 5 6 Consider a SISO system with a minimal model where ẋ = Ax + Bu, y = Cx a(s) =det(si A) and C(sI A) B b(s) a(s) Then with u(t) = K lqr x(t), closed-loop dynamics are: ny det(si A + BK lqr )= (s p i ) where the p i ={ the left-hand-plane roots of (s)}, with (s) =a(s)a( s)+r b(s)b( s) i= Use this to find the optimal pole locations, and then use those to find the feedback gains required using acker. The pole locations can be found using standard root-locus tools. (s) =a(s)a( s)+r b(s)b( s) = + r G(s)G( s) = The plot is symmetric about the real and imaginary axes. Symmetric Root Locus 2n poles are plotted as a function of r The poles we pick are always the n in the LHP. Several leaps made here for now. We will come back to this LQR problem later.
Fall 2 6.3 5 7 LQR Notes. The state cost was written using the output y T y, but that does not need to be the case. Wearefreetodefine a new system output z = C z x that is not based on a physical sensor measurement. Z J LQR = x T (t)(cz T C z )x(t)+ru(t) 2 dt Selection of z used to isolate the system states you are most concerned about, and thus would like to be regulated to zero. 2. Note what happens as r ; high control cost case a(s)a( s)+r b(s)b( s) = a(s)a( s) = So the n closed-loop poles are: Stable roots of the open-loop system (already in the LHP.) Reflection about the jω-axis of the unstable open-loop poles. 3. Note what happens as r ; low control cost case a(s)a( s)+r b(s)b( s) = b(s)b( s) = Assume order of b(s)b( s) is2m<2n So the n closed-loop poles go to: The m finite zeros of the system that are in the LHP (or the reflections of the systems zeros in the RHP). The system zeros at infinity (there are n m of these).
Fall 2 6.3 5 8 Note that the poles tending to infinity do so along very specificpaths so that they form a Butterworth Pattern: At high frequency we can ignore all but the highest powers of s in the expression for (s) = (s) = ; ( ) n s 2n + r ( ) m (b o s m ) 2 = s 2(n m) =( ) n m+b2 o r The 2(n m) solutions of this expression lie on a circle of radius (b 2 /r) /2(n m) at the intersection of the radial lines with phase from the negative real axis: ± lπ n m, l =,,...,n m 2, (n m) odd (l + /2)π ± n m Examples:, l =,,...,n m 2, n m Phase 2 ±π/4 3, ±π/3 4 ±π/8, ±3π/8 (n m) even Note: Plot the SRL using the 8 o rules (normal) if n m is even and the o rules if n m is odd.
Fall 2 6.3 5 9 5 Figure : Example#: G(s) = 8 4 2 (s+8)(s+4)(s+2) Symmetric root locus 4 3 2 Imag Axis 2 3 4.2 5 3 2 2 3 Real Axis Step Response Closed loop Freq Response.8 3.366 Y output.6 2.42 2.948 5.676 G cl.4 2.2..2.3.4.5.6.7.8.9 time (sec) 3 2 Freq (rad/sec)
Fall 2 6.3 5 Figure 2: Example #2: G(s) =.94 s 2.297 Symmetric root locus.8.6.4.2 Imag Axis.2.4.6.8.2.8.6.4.2.2.4.6.8 Real Axis Step Response Closed loop Freq Response.8.446 Y output.6.6489.3594 G cl.4 2.2 5 5 2 25 3 time (sec) 3 3 2 Freq (rad/sec)
Fall 2 6.3 5 5 Figure 3: Example #3: G(s) = 8 4 2 (s 8)(s 4)(s 2) Symmetric root locus 4 3 2 Imag Axis 2 3 4.2 5 3 2 2 3 Real Axis Step Response Closed loop Freq Response.8 3.366 Y output.6 23.42 2.9479 9.44262 G cl.4 2.2..2.3.4.5.6.7.8.9 time (sec) 3 2 Freq (rad/sec)
Fall 2 6.3 5 2 Figure 4: Example #4: G(s) = Symmetric root locus (s ) (s+)(s 3).8.6.4.2 Imag Axis.2.4.6.8.8.6 5 4 3 2 2 3 4 5 Real Axis Unstable, NMP system Step Response 4.3589 7.3589 3.6794 Closed loop Freq Response.4 Y output.2 G cl.2.4.6.8 2 3 4 5 6 7 8 9 time (sec) 2 2 2 Freq (rad/sec)
Fall 2 6.3 5 3 8 Figure 5: Example #5: G(s) = Symmetric root locus (s 2)(s 4) (s )(s 3)(s 2 +.8s+4)s 2 6 4 2 Imag Axis 2 4 6.8.6.4 8 6 4 2 2 4 6 Real Axis Unstable, NMP system Step Response 3.623.869 4.64874 8.734 3.8 2 3 Closed loop Freq Response Y output.2.2.4 G cl 4 5 6.6.8 2 3 4 5 6 7 8 9 time (sec) 7 8 2 2 Freq (rad/sec)
Fall 2 6.3 5 4 As noted previously, we are free to pick the state weighting matrices C z to penalize the parts of the motion we are most concerned with. Simple example oscillator with x =[p, v] T A =, B = 2.5 but we choose two cases for z z = p = x and z = v = x 2 SRL with Position Penalty.5 SRL with Velocity Penalty.5.5.5 Imag Axis Imag Axis.5.5.5 2.8.6.4.2.2.4.6.8 Real Axis.5 2.5.5.5.5 2 Real Axis Figure 6: SRL with position (left) and velocity penalties (right) Clearly,choosingadifferent C z impacts the SRL because it completely changes the zero-structure for the system.
Fall 2 6.3 5 5 Summary Dominant second and prototype design approaches (Bessel and ITAE) place the closed-loop pole locations with no regard to the amount of control effort required. Designer must iterate on the selected bandwidth (ω n ) to ensure that the control effort is reasonable. LQR/SRL approach selects closed-loop poles that balance between system errors and the control effort. Easy design iteration using r poles move along the SRL. Sometimes difficult to relate the desired transient response to the LQR cost function. Nice thing about the LQR approach is that the designer is focused on system performance issues The pole locations are then supplied using the SRL.