Content. Professur für Steuerung, Regelung und Systemdynamik. Lecture: Vehicle Dynamics Tutor: T. Wey Date: , 20:11:52

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1 1 Content Overview 1. Basics on Signal Analysis 2. System Theory 3. Vehicle Dynamics Modeling 4. Active Chassis Control Systems 5. Signals & Systems 6. Statistical System Analysis 7. Filtering 8. Modeling, Simulation with Matlab/Simulink Date: , 20:11:52

2 2 Filtering 1. Observer Task 2. Observability 3. Luenberger Observer 4. Kalman-Bucy Filter Date: , 20:11:53

3 1. Observer Task 3 Observer Task Objective For several control tasks an output feedback is not sufficient. Additional state signals are needed for suitable control results. Often state signals are not measurable or high cost sensors should be avoided. Definitions An observer gives an estimate of (not directly measured) state signals x by means of the input signal u and available output signals y. If disturbances and noise are small, then this is an observer problem. With disturbances and noise it is called a filtering problem and an estimator has to be designed. Date: , 20:11:53

4 2. Observability 4 Linear System Observability A non-excited Σ LS (u(t)=0) is called complete observable in a closed interval [t 0, t e ], if for given t 0 and t e any state vector x(t 0 ) can be evaluated by knowledge of the output signal y(t) in [t 0, t e ]. Kalman criteria Rank P is equal to the dimension of the observable sub space. Only for Rank P equal to the system order n the system is complete observable. Date: , 20:11:53

5 2. Observability 5 Linear System Observability Mathematical evaluation of states x from inputs u and outputs y [ c T n 1] y t c T A x t =P x t =[ ] ẏ t c T A y n 1 t If matrix P is regular, x can be directly evaluated from y 1[ y t ẏ t x t =P y t ] n 1 Due to differentiation (noise!) this method is not suitable for system order n>3. Date: , 20:11:53

6 3. Luenberger Observer 6 Basic observer scheme Inputs Plant excitation Plant output (all measured signals) (Error signal between plant output and model output) Outputs Estimate of plant output Estimate for state space vector Mathematical model Needs to be as exact as possible Real time usability No differentiation necessary for evaluation of x, only integration. Date: , 20:11:54

7 3. Luenberger Observer Example CDC - Continuous Damping Control Inputs Road excitation (difficult to measure) Body acc. (sensor based) (Error signal between body acc. and estimated body acc.) Output Estimate of relative speed between wheel and body Estimate of body acc. and wheel hop Road input CDC x CDC Model x^ - e(t) Observer 7 Body acc. Date: , 20:11:54

8 3. Luenberger Observer 8 Luenberger Observer: MIMO case Linear plant model Linear observer model Plant assumed to be non-observable (non-observable state(s) exist) u(t) B(t) B Non observable ẋ t x(t 0 ) I A(k) n A x(t)=? C y(t) Starting condition x(t 0 ) unknown x t 0 K e(t) - Constant feedback matrix K B x t IA(k) n C Dynamic of observer tunable ė t =[ A K A ]e t A x t Date: , 20:11:55

9 4. Kalman-Bucy-Filter 9 Basic scheme Kalman-Bucy-Filter Closed control loop Objective: for a given dynamic system in state space form is an optimal estimation of the state space vector x(t) to be found. Given: as measured signals we have the inputs and outputs of the system, which may be overlayed by measuring errors (bias, noise). Date: , 20:11:55

10 4. Kalman-Bucy-Filter 10 Kalman-Bucy Filter for linear plant Time-varying system Parameter Noise n(t) u(t) B(t) n(t) ẋ t x(t 0 ) IA(k) n C(t) m(t) Measurement error m(t) Variable feedback K(t) Non observable A(t) x(t)=? x t 0 =E {x t 0 } K(t) e(t) - Estimation error B(t) x t IA(k) n C(t) e t =x t x t 0 A(t) x t Date: , 20:11:55

11 4. Kalman-Bucy-Filter Matrix-Ricatti-equation Estimate and feedback matrix Unbiased estimate of minimum variance for x(t) x t =A t x t B t u t K t y t C t x t ; 11 with x t 0 =E {x t 0 } For K(t) we have: K t = P t C T t R 1 t with Matrix-Ricatti-equation P t =A t P t P t A T t P t C T t R 1 t C t P t Q t R t =E {w t w T t }; Q t =E {v t v T t } Date: , 20:11:56

12 4. Kalman-Bucy-Filter Kalman-Bucy Filter Example: Sensor calibration Time constant systematic fraction x(t) (bias) Time uncorrelated stochastic signal fraction m(t) (white noise) Objective: Sensor calibration No wanted signal (=0) Observe output signal Sensor x(t) =3 Abtrieb links m(t) =2 12 y(t) x t =K t [y t x t ] - Estimation of measuring error x t A(k) I n ẋ t K(t) Date: , 20:11:56

13 4. Kalman-Bucy-Filter Kalman-Bucy Filter Example continued... x t I n A(k) Sensor output y(t) ẋ t - K(t) 13 Requirements Covariance R t =E {m 2 t }=4 Systematic error P t 0 =E {x 2 t 0 }=9 Error covariance: P t = P 2 t 4 P t = t Feedback matrix K t = P t C T t R 1 t = 9 4 9t Date: , 20:11:56

14 4. Kalman-Bucy-Filter 14 Kalman-Bucy Filter Systematic error: value 2 White noise: variance 2 Kraft Modellbildung wird zur in Seite Matlab mit dem geringsten Widerstand verteilt ein durchdrehendes Rad w(t) möglich 2 x(t) 2 Estimates of measuring error Antrieb K(t) = 0,2 K(t) = 1,0 K(t) = 9/(4+9*t) t/s 10 Schätzwert x(t) 1 s Integrator 9/(4+9*u) K(t) t Date: , 20:11:57

15 Matlab/Simulink Exercise 15 Observer for vertical dynamics A QVM as described in fig. 1 is the plant to be estimated. Available input is road profile and available output is height between wheel and body. Internally disturbances are acting on in- and output of the system. Parameters are c 1 = 150kN/m c 2 = 15000N/m m 1 = 30kg m 2 = 300kg d = 1400kg/s fig. 1 Find an estimate for the force acting on the body (=body acceleration x m 2 ) For CDC (continuous damping control) the relative speed between wheel and body is needed. Find an estimate value for it. In general, road profile is not available for measurement. How does the estimate change under this condition? Date: , 20:11:57

Introduction to Engineering System Dynamics

CHAPTER 0 Introduction to Engineering System Dynamics 0.1 INTRODUCTION The objective of an engineering analysis of a dynamic system is prediction of its behaviour or performance. Real dynamic systems are