Topic #18 16.31 Feedback Control Closed-loop system analysis ● Robustness State-space -eigenvalue analysis Frequency domain- Nyquist theorem Sensitivity Copyright 2001 by Jonathan How
Topic #18 16.31 Feedback Control Closed-loop system analysis • Robustness • State-space — eigenvalue analysis • Frequency domain — Nyquist theorem. • Sensitivity Copyright 2001 by Jonathan How. 1
Fal!2001 16.3118-1 Combined estimators and requlators mal regulator to design the controller, the compensator is called A When we use the combination of an optimal estimator and an op Linear Quadratic Gaussian(LQG) Special case of the controllers that can be designed using the separation principle The great news about an lqg design is that stability of the closed loop system is guaranteed The designer is freed from having to perform any detailed me- chanics- the entire process is fast and can be automated Now the designer just focuses on How to specify the state cost function (i.e. selecting z= C2 a) and what value of r to use Determine how the process and sensor noise enter into the system and what their relative sizes are(i.e. select Ru ru So the designer can focus on the "performance"related issues, be- ing confident that the lQg design will produce a controller that stabilizes the system This sounds great-so what is the catch??
Fall 2001 16.31 18—1 Combined Estimators and Regulators • When we use the combination of an optimal estimator and an optimal regulator to design the controller, the compensator is called Linear Quadratic Gaussian (LQG) — Special case of the controllers that can be designed using the separation principle. • The great news about an LQG design is that stability of the closedloop system is guaranteed. — The designer is freed from having to perform any detailed mechanics - the entire process is fast and can be automated. • Now the designer just focuses on: — How to specify the state cost function (i.e. selecting z = Czx) and what value of r to use. — Determine how the process and sensor noise enter into the system and what their relative sizes are (i.e. select Rw & Rv) • So the designer can focus on the “performance” related issues, being confident that the LQG design will produce a controller that stabilizes the system. • This sounds great — so what is the catch??
Fal!2001 16.3118-2 The remaining issue is that sometimes the controllers designed using these state-space tools are very sensitive to errors in the knowledge of the model i. e. Might work very well if the plant gain a= 1, but be unstable if it is a=0.9 or a=1.1 LQG is also prone to plant-pole/compensator-zero cancellation which tends to be sensitive to modeling errors The good news is that the state-space techniques will give you a controller very easily You should use the time saved to verify that the one you designed is a"good"controller . There are of course. different definitions of what makes a controller good, but one important criterion is whether there is a reason able chance that it would work on the real system as well as it does in matlab → Robustness The controller must be able to tolerate some modeling error because our models in Matlab are typically inaccurate 3 Linearized model 3 Some parameters poorly known 3 ignores some higher frequency dynamics Need to develop tools that will give us some insight on how well a controller can tolerate modeling errors
Fall 2001 16.31 18—2 • The remaining issue is that sometimes the controllers designed using these state-space tools are very sensitive to errors in the knowledge of the model. — i.e., Might work very well if the plant gain α = 1, but be unstable if it is α = 0.9 or α = 1.1. — LQG is also prone to plant—pole/compensator—zero cancellation, which tends to be sensitive to modeling errors. • The good news is that the state-space techniques will give you a controller very easily. — You should use the time saved to verify that the one you designed is a “good” controller. • There are, of course, different definitions of what makes a controller good, but one important criterion is whether there is a reasonable chance that it would work on the real system as well as it does in Matlab. ⇒ Robustness. — The controller must be able to tolerate some modeling error, because our models in Matlab are typically inaccurate. 3 Linearized model 3 Some parameters poorly known 3 Ignores some higher frequency dynamics • Need to develop tools that will give us some insight on how well a controller can tolerate modeling errors
Fal!2001 16.3118-3 Example ● Consider the“( art on a stick” system, with the dynamics as given in the notes on the web define Then with y=a Ax+ Bu Cx For the parameters given in the notes, the system has an unstable pole at +5.6 and one at s=0. There are plant zeros at +5 The target locations for the poles were determined using the SrL for both the regulator and estimator Assumes that the process noise enters through the actuators Bw= b, which is a useful approximation Regulator and estimator have the same srL Choose the process/sensor ratio to be r/10 so that the estimator poles are faster than the regulator ones The resulting compensator is unstable(+16 But this was expected.(why?
Fall 2001 16.31 18—3 Example • Consider the “cart on a stick” system, with the dynamics as given in the notes on the web. Define q = ∙ θ x ¸ , x = ∙ q q˙ ¸ Then with y = x x˙ = Ax + Bu y = Cx • For the parameters given in the notes, the system has an unstable pole at +5.6 and one at s = 0. There are plant zeros at ±5. • The target locations for the poles were determined using the SRL for both the regulator and estimator. — Assumes that the process noise enters through the actuators Bw ≡ B, which is a useful approximation. — Regulator and estimator have the same SRL. — Choose the process/sensor ratio to be r/10 so that the estimator poles are faster than the regulator ones. • The resulting compensator is unstable (+16!!) — But this was expected. (why?)
Fal!2001 16.3118-4 Symmetric root locus Real Axis be gure 1: SRL for the regulator and estimator Fi Freq(rad/sec) Figure 2: Plant and Controller
Fall 2001 16.31 18—4 −8 −6 −4 −2 0 2 4 6 8 −10 −8 −6 −4 −2 0 2 4 6 8 10 Real Axis Imag Axis Symmetric root locus Figure 1: SRL for the regulator and estimator. 10−2 10−1 100 101 102 10−4 10−2 100 102 104 Freq (rad/sec) Mag Plant G Compensator Gc 10−2 10−1 100 101 102 0 50 100 150 200 Freq (rad/sec) Phase (deg) Plant G Compensator Gc Figure 2: Plant and Controller
Fal!2001 16.3118-5 Freq(rad/sec) Figure 3: Loop and margins Looking at both the Loop plots and the root locus, this system is stable with a gain of 1, but Unstable for a gain of 1+e and or a slight change in the system (possibly due to some unmodeled delays Very limited chance that this would work on the real system Of course, this is an extreme example and not all systems are like this, but you must analyze to determine what robustness mar gins your controller really ha Question what analysis tools should we use?
Fall 2001 16.31 18—5 10−2 10−1 100 101 102 10−2 10−1 100 101 Freq (rad/sec) Mag Loop L 10−2 10−1 100 101 102 −300 −250 −200 −150 −100 Freq (rad/sec) Phase (deg) Figure 3: Loop and Margins • Looking at both the Loop plots and the root locus, this system is stable with a gain of 1, but — Unstable for a gain of 1±² and/or a slight change in the system phase (possibly due to some unmodeled delays) — Very limited chance that this would work on the real system. • Of course, this is an extreme example and not all systems are like this, but you must analyze to determine what robustness margins your controller really has. • Question: what analysis tools should we use?
Fal!2001 16.3118-6 Real Axis Figure 4: Root Locus with frozen compensator dynamics. Shows ser sitivity to overall gain-symbols are a gain of 0.995: 0001: 1.005 0.5 Real Axis
Fall 2001 16.31 18—6 −10 −8 −6 −4 −2 0 2 4 6 8 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 Real Axis Imag Axis Figure 4: Root Locus with frozen compensator dynamics. Shows sensitivity to overall gain — symbols are a gain of [0.995:.0001:1.005]. −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Real Axis Imag Axis
Fal!2001 16.3118-7 Analysis Tools to∪se? Eigenvalues give a definite answer on the stability(or not) of the closed-loop system Problem is that it is very hard to predict where the closed-loop poles will go as a function of errors in the plant model Consider the case were the model of the system is Aoe+ Bu Controller also based on Ao, so nominal closed-loop dynamics A BK Ao- BK BK LC A0-BK-LC A0-LC But what if the actual system has dynamics =(A+△Ax+Bu Then perturbed closed-loop system dynamics are A0+△A BK Ao+△A-BKBK LC A0-BK-LC/ △A A0-LC Transformed Acl not upper-block triangular, so perturbed closed- loop eigenvalues are NoT the union of regulator estimator poles Can find the closed-loop poles for a specific AA, but Hard to predict change in location of closed-loop poles for a range of possible modeling errors p
Fall 2001 16.31 18—7 Analysis Tools to Use? • Eigenvalues give a definite answer on the stability (or not) of the closed-loop system. — Problem is that it is very hard to predict where the closed-loop poles will go as a function of errors in the plant model. • Consider the case were the model of the system is x˙ = A0x + Bu — Controller also based on A0, so nominal closed-loop dynamics: ∙ A0 −BK LC A0 − BK − LC ¸ ⇒ ∙ A0 − BK BK 0 A0 − LC ¸ • But what if the actual system has dynamics x˙ = (A0 + ∆A)x + Bu — Then perturbed closed-loop system dynamics are: ∙ A0 + ∆A −BK LC A0 − BK − LC ¸ ⇒ ∙ A0 + ∆A − BK BK ∆A A0 − LC ¸ • Transformed A¯cl not upper-block triangular, so perturbed closedloop eigenvalues are NOT the union of regulator & estimator poles. — Can find the closed-loop poles for a specific ∆A, but — Hard to predict change in location of closed-loop poles for a range of possible modeling errors
Fal!2001 16.3118-8 Frequency Domain Tests Frequency domain stability tests provide further insights on the " stability margins Recall from the Nyquist Stability Theorem P=# poles of L(s)=G(s Gc(s)in the RHP Z=# closed-loop poles in the RHP clockwise encirclements of the Nyquist Diagram about he critical point-1 Can show that Z=N+P(see notes on the web) So for the closed-loop system to be stable, need If the loop transfer function L(s)has P poles in the RhP s-plane (and lims-ooL(s)is a constant), then for closed-loop stability, the locus of l(ju)foru∈(-∞,∞) must encircle the critical point (1,0) P times in the counterclockwise direction Ogata 528 This provides a binary measure of stability, or not
Fall 2001 16.31 18—8 Frequency Domain Tests • Frequency domain stability tests provide further insights on the “stability margins”. • Recall from the Nyquist Stability Theorem: — P = # poles of L(s) = G(s)Gc(s) in the RHP — Z = # closed-loop poles in the RHP — N = # clockwise encirclements of the Nyquist Diagram about the critical point -1. Can show that Z = N + P (see notes on the web). So for the closed-loop system to be stable, need Z , 0 ⇒ N = −P • If the loop transfer function L(s) has P poles in the RHP s-plane (and lims→∞ L(s) is a constant), then for closed-loop stability, the locus of L(jω) for ω ∈ (−∞,∞) must encircle the critical point (-1,0) P times in the counterclockwise direction [Ogata 528]. — This provides a binary measure of stability, or not
Fal!2001 16.3118-9 ● Can use“ closeness”ofL(s) to the critical point as a measure of closeness"to changing the number of encirclements Premise is that the system is stable for the nominal system has the right number of encirclements e goal of the robustness test is to see if the possible perturbations to our system model(due to modeling errors)can change the number of encirclements In this case, say that the perturbations can destabilize the system Nichols: U Phase(deg) Figure 5: Nichols Plot for the cart example which clearly shows the sensitivity to the overall gain and or phase lag
Fall 2001 16.31 18—9 • Can use “closeness” of L(s) to the critical point as a measure of “closeness” to changing the number of encirclements. — Premise is that the system is stable for the nominal system ⇒ has the right number of encirclements. • Goal of the robustness test is to see if the possible perturbations to our system model (due to modeling errors) can change the number of encirclements • In this case, say that the perturbations can destabilize the system. −260 −240 −220 −200 −180 −160 −140 −120 −100 10−1 100 101 Nichols: Unstable Open−loop System Mag Phase (deg) −180.5 −180 −179.5 −179 −178.5 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 Nichols: Unstable Open−loop System Mag Phase (deg) 1 0.99 1.01 Figure 5: Nichols Plot for the cart example which clearly shows the sensitivity to the overall gain and/or phase lag
