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