Uncertain Systems. Robust vs. Adaptive Control. Generic Dynamic System ( Plant )
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1 Uncertain Systems Stochastic Robustness of Control Systems Robert Stengel Princeton University May 2009 Robustness Monte Carlo Evaluation Confidence Interval vs. Number of Trials Robustness Benchmark Problem Design of LTI and NTV Control Systems!The problem: physical plants have uncertain! Initial conditions! Inputs! Measurements! Dynamic structure! System parameters!design goal: control system that provides satisfactory plant response in the presence of uncertainty Copyright 2009 by Robert Stengel. All rights reserved. Generic Dynamic System ( Plant Robust vs. Adaptive Control! robust (OED!... statistical test that yields approximately correct results despite the falsity of assumptions!... result is largely independent of certain aspects of the input! Robust controller has fixed parameters that minimize response sensitivity to uncertain conditions! Uncertainties! Initial conditions! Inputs! Measurements! Plant structure! Plant parameters! Remediation! Feedback control! Feedback control! State Estimation! Robustness or adaptation (or both! adaptive (OED! a form of control in which the control parameters are automatically adjusted as conditions change so as to optimize performance! Adaptive controller measures change in conditions and adjusts control-system parameters and/or structure accordingly
2 Example of a Nonlinear, Time-Varying Adaptive Controller (Ferrari Classical Single-Input/Single-Output (SISO Control System Design! Approximate Dynamic Programming! On-line adaptive critic controller! Structure mimics good linear controller! Implements the nonlinear control law ( action network! Criticizes non-optimal performance via critic network! Adapts control gains to improve performance! Adapts cost model to improve estimate! Criteria for stability, performance, and robustness of a linear, time-invariant (LTI control system! Location of eigenvalues (roots of the closed-loop system! Envelopes of acceptable response in time and frequency domains! Satisfactory gain and phase margins s Plane SISO Control System Design Desirable Controller (Compensator Characteristics (Bode, 1934, 1945! Criteria for control system stability, performance, and robustness! Location of eigenvalues (roots of the closed-loop system! Envelopes of acceptable response in time and frequency domains! Satisfactory gain and phase margins (typically 6 db and 40!Make closed-loop response insensitive to! Plant parameter variations! Persistent disturbance inputs! Noisy measurement errors!this implies! Invariant response to inputs with low input frequency! Response roll-off at high input frequency! Smooth response transition at intermediate input frequency
3 Classical Robust Control System Design Properties of a PID Controller! Proportional-Integral-Derivative (PID control of a DC motor with rotary load! Forward-Loop Transfer Function y(s e(s = " c + c s + c % I P D s2 " 1 % $ ' # s &# $ Js 2 & ' = " c + c s + c % I P D $ s2 # Js 3 ' &! Closed-Loop Transfer Function y(s y C (s = y(s e(s 1+ y(s e(s [ ] = c I + c P s + c D s 2 c I + c P s + c D s 2 + Js 3! PID Control Law (or compensator e(t = y C (t " y(t u(t = c I de(t # e(t dt + c P e(t + c D dt! Frequency response (substitute j! for s! As! -> 0 y( j" y C ( j" # c I c I =1! As! ->! y( j" y C ( j" # $c D" 2 $ jj" 3 = c D jj" = $ jc D J" AR # c D ; % # $90 deg J" See for Ziegel-Nichols Tuning Method Linear, Time-Invariant System Equations Transfer Function Matrix! Time-Domain x (t = F x(t + Gu(t y(t = H x x(t + H u u(t! Frequency Domain (Laplace Transform sx(s " x(0 = F x(s + Gu(s y(s = H x x(s + H u u(s Dynamic Equation Output Equation Dynamic Equation Output Equation! Laplace Transform of Output Vector y(s = H x x(s + H u u(s = H x si " F = [ H x ( si " F "1 G + H u ]u(s + H x si " F [ ] "1 [ x(0 + Gu(s ] + H u u(s [ ] "1 x(0 = Control Effect + Initial Condition Effect! Transfer Function Matrix relates control input to system output! with H u = 0 and neglecting initial condition [ ] u(s = % H x y(s = H x ( si " F "1 G # $ ( si " F Adj si " F & G( u(s '
4 Expressions of Eigenvalue Uncertainty Stochastic Root Loci for Second-Order Example! Characteristic equation of the system! Uncertainty may be expressed as variation in! Elements of F! Coefficients of!(s! Eigenvalues! Frequency response/singular values/time response! Variation may be! Deterministic, e.g.,! Upper/lower bounds! Probabilistic, e.g.,! Gaussian distribution! Bounded variation is equivalent to probabilistic variation with uniform distribution si " F = det( si " F # $(s = s n + a n"1 s n" a 1 s + a 0 ( ( s " % 2 (...( s " % n = 0 = s " % 1 Uniform Distribution Gaussian Distribution! Nonlinear mapping from probability density functions (pdf of uncertain parameters to pdf of roots! Finite probability of instability with Gaussian (unbounded distribution! Zero probability of instability for some uniform distributions The Intractability of Uncertainty Breaking the Intractability of Uncertainty!Designing robust control systems for plants with real parameter variations is computationally intractable!number of possible plant models = m n, where!m = Resolution of parameter variations!n = Number of parameters!structured singular-value analysis for LTI systems is NP-Hard (Demmel; Newlin and Young; others, 1992!Three Solution Approaches!Exact Solution of Approximate Model! Deterministic bounds, e.g., µ synthesis! Limited number of uncertain parameters!approximate Solution of Approximate Model! e.g., µ synthesis with search! Limited number of uncertain parameters!approximate Solution of "Exact" Model! Probabilistic methods, e.g., Monte Carlo evaluation and search! Unlimited number of uncertain parameters
5 Probabilistic Control Design Estimating the Probability of Coin Flips! Design constant-parameter controller (CPC for satisfactory stability and performance in an uncertain environment! robust (OED:... statistical test... approximately correct... despite the falsity of certain... assumptions;... result is largely independent of certain aspects of the input.! Monte Carlo Evaluation of simulated system response with! competing CPC designs! given statistical model of uncertainty in the plant! Search for best CPC! Exhaustive search! Random search! Multivariate line search! Genetic algorithm! Simulated annealing! Single coin! Exhaustive search: Correct answer in 2 trials! Random search (20,000 trials! 21 coins! Exhaustive search: Correct answer in m n = 2 21 = 2,097,152 trials! Random search (20,000 trials... % Single coin x = []; prob = round(rand; for k = 1:20000 prob = round(rand * (1/(k prob * (k/(k+1; x = [x prob]; end figure plot(x, grid % 21 coins y = []; prob = round(rand; for k = 1:20000 for j = 1:21 coin(j = round(rand; end score = sum(coin; if score > 10 result = 1; else result = 0; end prob = result * (1/(k prob * (k/(k+1; y = [y prob]; end figure plot(y, grid Random Search Excels When There are Many Uncertain Parameters Binomial Distribution! Single coin! Exhaustive search: Correct answer in 2 trials! Random search (20,000 trials! 21 coins! Exhaustive search: Correct answer in m n = 2 21 = 2,097,152 trials! Random search (20,000 trials! Outcome of coin flip follows a binomial distribution! Confidence intervals of probability estimate are functions of! Actual probability! Number of trials Maximum Entropy when Pr = 0.5
6 ! Definition of design elements! Given a specific instantiation of the plant and the controller,! Response of System {C(d j,h(v i } H(v RNG : C(d j : v RNG : d j :! Is stable or not! Satisfies performance criteria or does not Plant H(v : C(d : v : pr(v : d : Controller Control System Design Problem Plant Controller Vector of uncertain parameters Probability density function vector of v Vector of control design parameters Uncertain parameter vector from random number generator Control design vector produced by search! Evaluation criteria are like coin flips, i.e., they follow binomial distributions! Probability distribution of the outcome is independent of! The number of uncertain parameters! Their probability distributions X-29 Aircraft Example of an Uncertain LTI Plant (Ray! Longitudinal dynamics for a Forward- Swept-Wing Demonstrator x T = ( V,",q,# ; u T = ( $E,$T $ "2gf 11 /V #V 2 f 12 /2 #Vf 13 "g' $ g 11 g 12 ' & & "45 /V 2 #Vf F = & 22 /2 1 0 ; G = & 0 0 & 0 #V 2 f 32 /2 #Vf 33 0 & g 31 g 32 & & % ( % 0 0 (! Nominal eigenvalues (one unstable " 1#4 = #0.1± j, # 5.15, 3.35! " and V have uniform distributions (±30%! 10 Aerodynamic coefficients have Gaussian distributions (# = 30% p = [" V f 11 f 12 f 13 f 22 f 32 f 33 g 11 g 12 g 31 g 32 ] Environment Uncontrolled Dynamics Control Effect Linear-Quadratic Regulators for the FSW Example Estimates of Probability of Instability for Three Linear-Quadratic Regulators! Cost function and optimal control law! Case a LQR with low control weighting! Case b LQR with high control weighting! Case c Case b with gains multiplied by 5 for bandwidth recovery minj = 1 u 2 " # ( x T Qx + u T Rudt; u = $Cx 0 Q = diag( 1,1,1,0 ; R = ( 1,1 ; " 1#4 nominal = #35,#5.1,#3.3,#.02 $ ' C = & % 0.98 #11 #3 #1.1( Q = diag( 1,1,1,0 ; R = ( 1000,1000; " 1#4 nominal = #5.2,#3.4,#1.1,#.02 $ #0.06' C = & % 0.01 #63 #16 #1.9 ( " 1#4 nominal = #32,#5.2,#3.4,#0.01 $ #0.32 ' C = & % 0.05 #313 #81 #1.1# 9.5( with Gaussian Aerodynamic Uncertainty
7 Stochastic Root Loci for Three Linear-Quadratic Regulators Effects of Parameter Distributions and Control Weight on the Probability of Instability Case a: Low Control Weights with Gaussian Aerodynamic Uncertainty! Magnitude of control weight varied by $ minj = 1 u 2 # $ ( x T Qx + u T "Rudt; u = %Cx 0 Case b: High Control Weights Case c: Bandwidth Recovery! Probabilities of instability with 30% uniform aerodynamic uncertainty! Case a: 3.4 x 10-4! Case b: 0! Case c: 0! Too much control! Control effect uncertainty dominates! Too litle control! Static instability uncertainty dominates Probability of Not Satisfying a Design Metric Optimizing Control System Robustness! For a given design vector, d j, a design metric, m, does or does not satisfy a criterion where # m = $ 0 : % 1: V = Pr(satisfactory + Pr(not satisfactory = 1 Pr S ( d j + Pr NS d j ( =1! The probability that the control design does not satisfy the criterion equals the expected value of the metric Pr NS (d j,m = E m C( d j,h v [ (,H( v ] pr( vdv { [ ( ]} = m C d j Criterion is satisfied " V Criterion is not satisfied Space of all possible v! Minimize the probability of not satisfying a criterion with respect to the design vector, d j min Pr NS (d,m d where [ ] = min N = n NS = d { m[ C( d,h ( v ]pr( vdv} " V & 1 = min' lim d N #$ ( N N % i=1 [ ( ] m C( d,h v RNG & n = min lim NS ' * d ( N #$ N +, min & n ' NS d ( N Number of trials * + * + Number of trials that are not satisfactory
8 1990 American Control Conference Robust Control Benchmark Problem Nominal Plant and Contest Goals u m 1 m 2! Challenge: Design a feedback compensator for a 4 th -order spring-mass system ( the plant whose parameters are bounded but unknown! Minimize the likelihood of instability! Satisfy a settling time requirement! Don t use too much control k y! Transfer function! Nominal eigenvalues ( k /m 1 m 2 H uy (s = " s 2 s 2 + k(m + m % 1 2 $ ' # m 1 m 2 & " 1#4 = ± j k(m 1 + m 2 m 1 m 2, 0, 0! Contest goals! Case 1: 15-s settling time (I.C. response for nominal massspring values (= 1 and closed-loop stability for 0.5 < k < 2! Case 2: Similar goals with sinusoidal disturbance! Case 3: Uncertain m 1, m 2, and k with unspecified bounds! Minimal control effort and complexity Proposed Control Design Approaches Nominal Step Responses of Design-Entry Compensators!Contest Entries!Approximate loop transfer recovery (A, B, C!H! design using bilinear transform (D!Optimization using nonlinear programming (E!LQG regulator with structured covariance (F!Game theoretic controller (G!H! design using internal model principle (H, I, J!Design-entry compensators had!2 to 8 eigenvalues!2 to 5 zeros A B, C E F G J H I D
9 Probability of Instability vs. Gain Margin of A-J Probability of Instability vs. Phase Margin of A-J Probability of Settling Time Exceedance of A-J Probability of Excess Control Usage of A-J
10 Optimal Stochastic Robustness Problem Design Approaches to Optimize Stochastic Robustness! Parameters of mass-spring system! Uniform probability density functions for! 0.5 < m 1, m 2 < 1.5! 0.5 < k < 2! Probability of Instability, P i! m i = 1 (unstable or 0 (stable! Probability of Settling Time Exceedance, P ts! m ts = 1 (exceeded or 0 (not exceeded! Probability of Control Limit Exceedance, P u! m u = 1 (exceeded or 0 (not exceeded! Design Cost Function! J = a P i 2 + b P ts 2 + c P u 2 a = 1, b = c = 0.01! SISO Linear-Quadratic-Gaussian Regulators (Marrison! Implicit model following with control-rate weighting and scalar output (5 th order! Kalman filter with single measurement (4 th order! Design parameters: control cost function weights, springs and masses in ideal model, estimator weights! Search: Multivariate line search and genetic algorithm! Sweep and Search SISO Compensators with Genetic Algorithm (Wang! Family of low-order compensators! Design parameters: numerator and denominator coefficients! Search: Genetic algorithm Comparison of Design Costs with LQG Controllers Search-and-Sweep Design of Family of Robust Feedback Compensators Cost Emphasizes Instability Cost Emphasizes Excess Control Cost Emphasizes Settling- Time Exceedance 1 Begin with lowest-order feedback compensator a C 12 (s = 0 + a 1 s b 0 + b 1 s + b 2 s " C d 2 ( 2 Arrange parameters as binary design vector d = { a 0,a 1,b 0,b 1,b 2 } 3 Genetic algorithm search for best values of the design vector, i.e., design vector that minimizes J d* = { a 0 *,a 1 *,b 0 *,b 1 *,b 2 *}
11 Genetic Algorithm Search-and-Sweep Design of Family of Robust Feedback Compensators! Randomly select initial population of design vectors ( chromosomes! Use Monte Carlo Evaluation to estimate design cost for each design vector! Randomly crossover tails of design pairs! Mutate on rare occasions! Reproduce population for next generation 1 Define next higher-order compensator C 22 (s = a 0 + a 1 s + a 2 s2 b 0 + b 1 s + b 2 s 2 2 Optimize over all parameters, including optimal coefficients in starting population d = { a 0 *,a 1 *,a 2,b 0 *,b 1 *,b 2 *} " d** = { a 0 **,a 1 **,a 2 **,b 0 **,b 1 **,b 2 **} 3 Sweep to satisfactory design or no further improvement C 23 (s = a 0 + a 1 s + a 2 s 2 b 0 + b 1 s + b 2 s 2 + b 3 s 3 C 33 (s = a 0 + a 1 s + a 2 s2 + a 3 s 3 b 0 + b 1 s + b 2 s 2 + b 3 s 3 C 34 (s = a 0 + a 1 s + a 2 s 2 + a 3 s 3 b 0 + b 1 s + b 2 s 2 + b 3 s 3 + b 4 s 4... Design Cost and Probabilities of Sweep-and-Search Compensators Effect of Number of Zeros on Design Cost of Optimal Sweep-and-Search Compensators! 2 nd - to 5 th -Order Proper Compensators Number of Zeros = Number of Poles! 2 nd - to 5 th -Order Proper Compensators nd Order 3rd Order 4th Order 5th Order Cost Prob Instab Prob Ts Viol x 0.01 Prob Cont Viol x 0.1
12 Estimate of Minimum"Design Cost from Eight Design Trials Robust PID Control Using Simulated Annealing (Motoda ALFLEX Test Vehicle Helicopter Drop Test PID Controller! 4 uncertain parameters! 4 control design parameters! Design metrics! Overshoot! Settling time! Tracking error Robust PID Control Using Simulated Annealing (Motoda * Linear and Nonlinear Robust Control of a Hypersonic Aircraft (Marrison, Wang Performance of Original and Optimized Controllers Comparison of Three Optimization Methods! Flight condition: Mach 15 at altitude of 115,000 ft! 28 uncertain aerodynamic and thrust parameters! 39 design metrics! Linear-quadratic regulator! 10 control design parameters! Feedback linearization (nonlinear dynamic inversion + linear-quadratic regulator! 11 control design parameters * JGCD, Mar-Apr 2002,
13 Design Metrics for Longitudinal Control of a Hypersonic Aircraft (Marrison, Wang Comparison of Inverse-Dynamic and LQR Performance (Marrison, Wang! Closed-loop stability! 19 measures of velocity-command step response! 19 measures of altitude-command step response! 0.1% probability of instability for LQR (95% confidence! 0% probability of instability for Nonlinear Dynamic Inversion (95% confidence J-10 High-Incidence Research Model Control (GARTEUR Design Challenge (Wang Conclusions! Full flight envelope! 9 specified flight/maneuver conditions! 16 uncertain aerodynamic parameters Nichols Chart for NDI Design! SRAD NDI all-axis controller! 18 design metrics! 9 control design parameters Gain-Phase Exclusion Region Comparison! Intractability of robust control design! Stochastic robustness analysis and design! overcomes the intractability! provides analysis and design framework that is identical for linear and nonlinear systems! uses statistics of parameter/structural uncertainty that must be known for robust design! readily adapts to pre-existing design philosophies and tools! Produces control system designs that are demonstrably better than competing alternatives
14 References - 1 References - 2! Some Effects of Parameter Variations on the Lateral-Directional Stability of Aircraft, J. Guidance and Control, Vol. 3, No. 2, Mar-Apr 1980, pp (Also translated and republished in the Soviet journal, Rocket Technology and Cosmonautics, Vol. 19, No. 2, Feb 1981, pp ! Stochastic Robustness of Linear-Time-Invariant Control Systems, IEEE Trans. Automatic Control, Vol. 36, No. 1, Jan 1991, pp (with L.R. Ray.! Application of Stochastic Robustness to Aircraft Control Systems, J. Guidance, Control, and Dynamics, Vol. 14, No. 6, Nov-Dec 1991, pp (with L.R. Ray.! Robustness of Solutions to a Benchmark Control Problem, J. Guidance, Control, and Dynamics, Vol. 15, No. 5, Sept-Oct 1992, pp (with C. Marrison.! Stochastic Measures of Performance Robustness in Aircraft Control Systems, J. Guidance, Control, and Dynamics, Vol. 15, No. 6, Nov-Dec 1992, pp (with L.R. Ray.! A Monte Carlo Approach to the Analysis of Control System Robustness, Automatica, Vol. 29, No. 1, Jan 1993, pp (with L.R. Ray.! Computer-Aided Analysis of Linear Control System Robustness, Mechatronics, Vol. 3., No. 1, Jan. 1993, pp (with L.R. Ray.! Stochastic Robustness Synthesis Applied to a Benchmark Control Problem, Int'l. J. Robust and Nonlinear Control, Vol. 5, No. 1, Jan 1995, pp (with C. Marrison.! Probabilistic Evaluation of Control System Robustness, Int'l. J. of Systems Science, Vol. 26, No. 7, July 1995, pp (with L. R. Ray and C. Marrison.! Reply by Authors to J. J. Gribble (to comment on [4], J. Guidance, Control, and Dynamics, Vol. 18, No. 6, Nov-Dec 1995, pp (with C. Marrison.! Robust Control System Design Using Random Search and Genetic Algorithms, IEEE Trans. Automatic Control, Vol. 42, No. 6, June 1997, pp (with C. Marrison.! Design of Robust Control Systems for a Hypersonic Aircraft, J. Guidance, Control, and Dynamics, Vol. 21, No. 1, Jan-Feb 1998, pp (with C. Marrison.! Parallel Synthesis of Robust Control Systems, IEEE Trans. Control System Technology, Vol. 6, No. 6, Nov. 1998, pp (with W. Schubert.! Searching for Robust Minimal-Order Compensators, ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 123, No. 2, June 2001, pp (with Q. Wang.! Robust Control System Design Using Simulated Annealing, J. Guidance, Control, and Dynamics, Vol. 25, No. 2, Mar-Apr 2002, pp (with T. Motoda, and Y. Miyazawa.! Robust Control of Nonlinear Systems with Parametric Uncertainty, Automatica, Vol. 38, Sept 2002, pp (with Q. Wang.! Robust Nonlinear Flight Control of a High-Performance Aircraft, IEEE Trans. Control Systems Technology, Vol. 13, No. 1, Jan 2005, pp (with Q. Wang.! Probabilistic Control of Nonlinear Uncertain Dynamic Systems, Probabilistic and Randomized Methods for Design under Uncertainty, G. Calafiore and F. Dabbene, ed., Springer, New York, 2006, pp (with Q. Wang.
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