Control and system theory 9/5/28 State feedback, Observer, and State feedback using an observer KOSEKI, Takafumi The University of Tokyo
Fundamentals i What is a state observer? ii State feedback and classical output feedback iii Analogue observer: Differential operatopr and an observer iv Reduced observer v Disturbance observer vi Design of feedback gains: Kessler's and Manabe's canonical forms
Type of sensor Trend in motor drive ---Extent of Sensorless--- Speed FB Nothing needed Speed DSO Speed sensorless Speed/current sensorless Needed model information Dynamic equation and -order delay Internal motor dynamics Internal motor dynamics Sensor information Coarse position and current Voltage and current Current and motor in inverter
What is an observer? When no direct measurements are available? ==>Estimation from other measurable physical information based on plant model What is the ESTIMATION? Calculation based on knowledge of physical model? An example: T L DC motor drive Motor T m + + Js ω θ s Observer + + Js ωˆ θˆ s + - TˆL θ
Ex. Speed control of DC motor Assumptions i a i f are measured. I f const. is also directly measured. Information on angular speed ω shall be exactly acquired for direct feeedback Approach Is measured; Let's calculate the differential! Problem :.Realtime differential is NEVER possible: causality. 2.When Θ has noise --> The calculation is much noisier.
Approach 2 Ex. Control of DC-motor Let's calculate the integral of current information! T M =K i a i f = J T M T L = dt () (2) (3) The problem of the noise is solver, but... Problems :.() の K は正確か? Can we know the exact K in ()? 2. The load torque (2) is not measured. 3. We never know the exact amount of J. 4. What is the initial value of speed in (3)???
Ex. Control of DC-motor Problems in Approach 2 Uncertainty in modelling. No availability of the initial conditions: <=> observability Almost no information on load torque: <==shall be treated as DISTURBANCE Simulation solely does NOT work well! In Position sensorless controls: integral calculations as current =>torque =>angular acceleration ==> speed is NOT possible in principle.
Why don't we use both information? Approach 3 The ω in approach 2 is integrated furthermore and the calculated value of θ is obtained. If =, then the information of can be used for correction of the integral calculation! Let's check always the calculation by θ! Checking solution of the dynamic equation using OUTPUT signals ==> Estimation or Observation of not-directly measurable state variables.
State estimation in a DC-machine = K i f i a Correction l J using = l 2 This correction is possible in a computer. output Θ ( Here the meaning and design method of the gains l,l 2 have not been studied yet. ) Concerning = K i load f i a T torque: L is obvious. The basic idea of J J = derives J = T disturbance L observer.
Classical control and state feedback control Example: PD-controller ref + - K D s s K P PD-controller Plant y= The information of Θ and its approximate differential is used for feedback. ==>The controller itself has dynamics of differential. ( In the real world, you can use just APPROXIMATE differential operation. )
Classical control and state State feedback If state vector feedback control x= [ ] is directly measured, you need just multiply the constant gain matrix with the state vector No dynamics in FB-controller u - ω cannot be directly measured! --> Substitute the real ω by the estimated value obtained from the observer <State feedback + observer> system has the substantial dynamics in the observer side. B + + F ẋ x A x C y
Classical control and state feedback control State feedback REGULATOR x Stabilize at ZERO-point ẋ= A x B u u= F x Feedback rule u - ẋ= A BF x B + ẋ + A det {si A BF }= F l i m x= t An appropriate F, which enables to allocate all the poles in the left hand side of the s-domain, stabilize the system. x x C y
Plant Observer+ State-FB ẋ t = A x t B u t y t =C x t State equation Output equation In reality, x is not directly measured: the feedback og the x is impossible! Feedback rule u= F x x u, y
Full state observer Even if A, B, C-models completely identical to reality is available, we never know the initial state x t x t y t y t condition x O y t y t tells how wrong the estimation was. => Correct the estimation x t based on the info. Real world Basically the same In computer ẋ t = A x t B u t y t =C x t Correction by output error x t = A x t B u t y t =C x t
State feedback using the observer Read world ẋ t = A x t B u t y t =C x t Extended system Feedback rule d dt [ Model in computer Correction term x t = A x t B u t LC x x y t =C x t u= F x x ] x = [ A BF ] [ LC A LC BF x ] x Used for feedback control
Plant An example of observer feedback ẋ t = A x t B u t y t =C x t State vector is defined as t = t = K i f i J a J T L y= ] = [ ] [ ] [ d [ dt K i f x=[ ] J ] i a [, then J ] T L State equation Output equation State equation Output equation State equation y=[ ] [ ] Output equation: Output = Signals directly measured with sensors.
Dynamic behaviour in the example Dynamic equation of the observer x t = A x t B u t L y t y t = A LC x B u L y {s I A LC } X s x =BU s LY s lead X s ={s I A LC } x {s I A LC } B U s {s I A LC } LY s If we watch it concretely: s = l 2 s s s 2 s l K i f l s l 2 s 2 l s l 2 J I a s s l s 2 l s l 2 A low pass filter against noise in angle. When l 2 is large, the contribution of this term is dominant. ==> Approximate differential of angle. When l 2 is nearly zero, the contribution of these terms is dominant: Simple integral of angular acceleration.
Separation theorem Q: Design the feedback controller without any consideration of observer. After that design the observer. Is it really right? A: In usual, it is OK! Eigen polynomial of the extended system [ det {s I ext A ext }=det si A BF ] LC si A LC BF [ si A LC si A LC ] =det LC si A LC BF =det [ si A LC ] LC si A BF = si A LC si A BF Eigen polynomial of the observer Eigen polynomial of the controller The addition of the observer does not affect the controller pole place, if the modelling is good.
Reduced observer ( especially, minimal observer ) Q: The estimation of the measurable output signal in full-order observer is waste of time/computational power? A: It has sense, but the reduction of the number of estimated variables is possible Gopinath method Separation of measured variables from notmeasured ones to be estimated : An appropriate coordinates transformation is applied x= [ r ] Separation [ ṙ ] y ẏ = [ of variables A A 2 A 2 A 22 ] [ r y ] [ B ] u B 2
ṙ= A r A 2 y B u ẏ A 22 y B 2 u=a 2 r Known Minimal observer r x, A A, A 2 y B u B u Formal transformation ẏ A 22 y B 2 u y, A 2 C ẋ t = A x t B u t y t =C x t results in After estimating transformed vector by x t = A LC x B u L y then, original vector is r= A L A 2 r A 2 y B u L ẏ A 22 y B 2 u Differential variable in R.H.S is NOT good!
Minimal observer ( continued ) r= A L A 2 r A 2 y B u L ẏ A 22 y B 2 u Differential in R.H.S is not good. The introduction of P = r L y results in Ṗ= A L A 2 P B L B 2 u { A L A 2 L A 2 L A 2 2 } y After the estimation of P r=p L y shall be calculated.
Minimal observer in the DC-motor example Ṗ = A L A 2 P B L B 2 u { A 2 L A 2 L A 2 L A 22 } y Full order observer s = l 2 s s s 2 s l K i f l s l 2 s 2 l s l 2 J Minimal observer s = s l l K i f J s l I a s I a s s l s 2 l s l 2 s l s s Observer gain l =large Approximate differential θ time constant l=small Approximate integral of i a /l: When l is large, then the estimation is vulnerable to noise in θ
Stationary Kalmann filter and full-order observer The role of the observer When signals cannot be measured directly ==>To estimate them from other measurable physical values Mainly, the observer suppresses errors in/from initial values. If there are continuous and stochastic noises in process? ẋ= A x B u r t r t : System noise y=c x t t : Measurement noise Kalman filter suppresses harmful effects of noise using knowledge of the process model. ( Sensitive to r Insensitive to ρ is good filtering: estimator. )
Stationary Kalmann filter and full-order observer(continued) ẋ= A x B u r t : System noise y=c x t : Measurement noise Assumption : r t t stationary, averages are zero White Gaussian, and non-correlated ( If not, appropriate bias and filtering shall be applied. ) Q, S Correlation matrices are defined as follows. cov {r t,r t 2 }=E {r t r T t 2 }=Q t t 2 cov { t, t 2 }=E { t T t 2 }=S t t 2 Stationary Kalmann filter x= A LC x B u L y Determined through optimization of the square
Stationary Kalmann filter and full-order observer(continued) ẋ= A x B u r t : System noise y=c x t : Measurement noise Stationary Kalmann filter x= A LC x B u L y Determined through optimization of square norm Observer and Kalmann filter have identical structure The method for determining their gains is different: ( Kalman filter: Ricatti eq. gives the filter-gains. ) Large Q makes fast estimation. Large S makes slow estimation. Suppression of the effects of continuous stochastic disturbances
Disturbance observer Current I Motor constant K F L Load force Ms Plant dynamics ẋ s x K F L Estimated load force Ms s Approximate reversed dynamics F L = ゼロ次外乱の仮定 = 一種のゴミ箱のようなもの d [ dt =[ v L] F ] M [ v L] [ F K M ] I Zero-order disturbance observer = A garbage box
Digital observer u A/D 変換 B c + + s A c + + - B 2 C - z 2 + + A 2 L 2 C c n=,n θ A/D PG 変換 Observer: implemented in a digital processor Z-transform: dynamics represented in recurrence formula State transition matrix is often used for transformation from continuous form Deadbeat observes is possible, but has problems in practice. There are predictive and current observes
Controller design using polynomial method Practical design method in labs. Kessler's canonical form
Introduction Practical design controller/ observer gains for motor control and magnetic levitation in laboratories: plant identification, プラント同定 structure of controllers, 制御器構造 and gain tuning ゲイン設定 Ziegler-Nichols' method LQR(Optimal control)? Pole placement? Polynomial method : Denominator of a closed loop transfer function
Polynomial method C. Kessler's proposal Betragsopitimum Symmetrieopitimum Dämpfungopitimum (damping optimization) 96 at Siemens in Germany In Japan, Prof. Manabe modified the method to Manabe's Canonical form and mitigated the condition among coefficients in his coefficient diagram method base on his experience at Mitsubishi
Kessler's Canonial form a a n n n s + an s +... + as + a Ts + T = T = a a a T = 2 a a 2 2 T α = α 2 2 2 = = T2 a a2 T3 aa3 2 a Equivalent time constant The procedure of terms shall be reversed!,,,, α T n T = n n = = Tn a a n n a a 2 n n 2 a n α = α = α =... = α = 2 3 n Stability index 2. System time constant
Kessler's canonical form in general form a a n n n s + an s +... + as + a Ts + n s = n k= k k 2 T k s k
Response of Kessler's form () step response step response.2.8.6.4.2.5.5 2 2.5 3 3.5 4 4.5 5 time 2nd order Kessler form.2.8.6.4.2 st order Kessler form.5.5 2 2.5 3 3.5 4 4.5 5 time Step response of st and second order 3 2 - -2-3 -3-2 - Pole map: 2 nd order
Response of Kessler's form (2) step response step response.2.8.6.4.2.5.5 2 2.5 3 3.5 4 4.5 5 time 4th order Kessler form.2.8.6.4.2 3rd order Kessler form.5.5 2 2.5 3 3.5 4 4.5 5 time Step responses: 3 rd and 4 th order forms 2.5.5 -.5 - -.5-2 -3-2.5-2 -.5 - -.5 Pole map: 3 rd order form
Response of Kessler's form (3) step response step response.2.8.6.4.2 5th order Kessler form.5.5 2 2.5 3 3.5 4 4.5 5 time.2 6th order Kessler form.8.6.4.2.5.5 2 2.5 3 3.5 4 4.5 5 time 5 5-5 - -5-35 -3-25 -2-5 - -5 5 Step responses: 5th and 6 th order forms Pole map: 6 th order form
Response of Manabe's form step response step response.2.8.6.4.2.5.5 2 2.5 3 3.5 4 4.5 5 time 5th order Manabe form.2.8.6.4.2 5th order Kessler form.5.5 2 2.5 3 3.5 4 4.5 5 time Step responses: 5 th and 6 th order forms 5 5-5 - -5-35 -3-25 -2-5 - -5 5 Pole map: 6 th order form
Frequency responses of the nth-order Kessler's canonical form Bode Diagrams From: U() Phase (deg); Magnitude (db) To: Y() -5 - -5 - -2-3 -4-5 5 次 6 次 次 2 次 3 次 4 次 次 3 次 5 次 2 次 -6-2 4 次 6 次 Frequency (rad/sec) When the equivalent time constant T is identical.
Stability index and step responses ().6.4.2 Kessler form alpha=.5, 2., 2.5, 3,.5 2. 2.5 3. step response.8.6.4.2.5.5 2 2.5 3 3.5 4 4.5 5 time When the equivalent time constant is identical
Stability index and step responses (2) step response.6.4.2.8.6 Kessler form alpha=.5, 2., 2.5, 3,.5 2. 2.5 3..4.2 2 4 6 8 2 time When the system time constant Tsys is identical
Pole maps and stability index 8 6 6 4 2 4 2-2 -4-6 -2-4 -6-8 -4-2 - -8-6 -4-2 2-8 -6-4 -2 - -8-6 -4-2 2 5 th order α=.5 5 th order α=2. 4 2 3.5 2 - -2-3 -4 - -9-8 -7-6 -5-4 -3-2 -.5 -.5 - -.5-2 -7-6 -5-4 -3-2 - 5 th order α=2.5 5 th order α=3.
One dimensional maglev 位置制御器 ギャップ長指令 電圧源 制御電流源 制御巻線電流 電磁石 巻線 ギャップセンサ 浮上力 ギャップ長浮上鉄心 重力
State feedback stabilizing control ẋ t =A x t B u t u t = F x t
Additional integral term I-PD : PID Overshoot in step input response is mitigated. Kessler's canonical from; Proportional and differential gains in I-PD and PD controls are identical An additional state variable is used in state feedback
Introduction of an observer
Gain tuning in observer state feedback control. Extended system including controller and observer dynamics => 4+4=8 th order 2. Relative quickness of the observer 3. Independent designs of controller and observer
Summary of polynomial method Kessler's canonical form: Practical easy-to-use design method in laboratories Equivalent and system time constant: what should be kept constant? When the stability index is fixed, the gain tuning is single dimensional using the equivalent time constant Also useful for state feedback controllers and observers Mitigated gain condition: Manabe's coefficients' diagram Consideration of zero point location: Extended Damping Optimization
Summaries Fundamental : Comparison between classical and state feedbacks Structure and role of observers Observer and stationary Kalman filter Disturbance observer Gain tuning using polynomial method Fundamental discussion on digital observer Being important in industry application in future.