`State-space approach Contents: State space representation Pole placement by state feedback LQR (Linear Quadratic Regulator) Observer design Kalman Filter ­ LQG Separation Principle Spillover Frequency Shaped LQG HAC-LAC strategy(Ch.7, p.138)Transfer function approach:State variable form:State space Equation:Plant noise Measurement noise FeedthroughThe choice of state variable is not uniqueExample: s.d.o.f. oscillator: 1.ABAcceleration Output:C 2.D (feedthrough)A is dimensionally homogeneInverted PendulumEquation of motion: Linearization: Change of variable:with(natural frequency Of the pendulum)State variable form:Output equation:System transfer functions.d.o.f. oscillator:Inverted Pendulum:For SISO systems, one can write:Zeros polessuch that, for some initial condition, the free response is Free response:. Poles:are the eigenvalues of A, solution of2.Zeros:An input produces no output:applied from appropriate initial conditionsThe state vector has the form: If:That is if:Then: Y=0 if(1)(2)(1) And (2)dtm [ ] = 0Pole placement by state feedbackState feedbackIf the system is controllable, the closed-loop poles can be placed arbitrarily in the complex plane. The gain G can be chosen such thatClosed-loop characteristic equationSelected arbitrarilyExample: s.d.o.f. oscillator (1)Relocating the poles Deeper in le left-half planeExample: s.d.o.f. oscillator (2)State-space equation:State feedback: Closed-loop characteristic equation:Desired behaviour:Linear Quadratic Regulator (SISO)u such that the performance index J is minimizedControlled variable:Control force: u Weighing coefficientSolution: The closed-loop poles are the stable roots of:whereCharacteristic equation:Identical to that of:Weighing On the control-Symmetric root locus·Symmetric with respect to the imaginary axis As well as the real axis ·Only the roots in the left half plane have to be consideredExample: Inverted pendulum (1)Controlled Variable:Selected polesExample: Inverted pendulum (2) 1. Select the poles on the left side of the Symmetric root locus 2. Compute the gains so as to match the desired poles:Observer designFull stat observer (Luenberger observer):(perfect modeling !!)Duplicates the systemInnovation Error equation:Error:If the system is observable, the poles of the Error equation can be assigneg arbitrarily by Appropriate choice of ki In practice, the poles of the observer should Be taken 2 to 6 times faster than the regulator polesIn practice, there are modeling errors and measurement noise; These should be taken into account in selecting the observer gains One way to assign the observer poles: KALMAN filter (minimum variance observer) Scalar White noise processes The optimal poles location minimizing the variance of the Measurement error are the stable roots of the symmetric root locus:Whereis the T.F. between w and y and Plant noise intensity (w) a Measurement noise intensity (v)Example: Inverted pendulum (1)1. Assume that the noise enters the system at the input (E = B) proportional to The same root locus can be used for the regulator and the observer designExample: Inverted pendulum (2)2. AssumeObserver polesNote (SISO design) LQR =Controlled variable z Input uKalman filter =Output measurement y Plant noise wAssuming that z = y (H = C) and that the noise enters the plant at the input (E = BThe design of the regulator and the observer can be completed with the same Symmetric root locus corresponding to the open-loop transfer function G(s)Separation PrincipleClosed-loop equations:Reconstructed stateCompensator2n state variables WithBlock triangularthe eigenvalues are decoupledTransfer function of the compensatorThe poles of the compensator H(s) are solutions of the characteristic equation:·They have not been specified anywhere in the design ·They may be unstable ·H(s) is always of the same order as the systemThe two-mass problem (1) uX3X1 = yState-space equation:uLQG design with symmetric root locus based onTwo-mass problem (2): Symmetric root-locusOpen-loop polesTwo-mass problem (3) Design procedure:·Select the regulator poles on the locus ·Compute the corresponding gains G ·Select the observer poles (2 to 6 times faster) ·Compute the corresponding gains K ·Compute the compensator H(s)One finds:Notch filter !Two-mass problem (4): Root locus of the LQG controllerCompensatorNotch filterOptimum design for g= 1Two-mass problem (5): robustness analysis Effect of doubling the natural frequencyUnstable loop ! This frequency has been doubled The notch filter becomes uselessEffect of lowering the natural frequency by 20%Two-mass problem (6): Robustness analysisPole/zero Flipping ! The notch is unchanged(Ch.9, p.206)Spillover (1)Phase stabilizedCrossoverGain stabilized...BandwidthThe residual modes Near crossover may Be destabilized by SpilloverSpillover (2): mechanismFlexible structure dynamics Controlled modes Actuators Residual modes SensorsSpillover (3): EquationsStructure dynamics: Controlled modes: Residual modes: Output: Full state observer: Observation Spillover Control SpilloverFull state feedback:Spillover (4): Eigen value problemControl spilloverObservation spilloverIf either B r=0 or Cr=0, the eigen values remain decoupled If both Br and Cr exist, there is SpilloverSpillover (5): Closed-loop polesThe residual modes have a small stability margin (damping !) and can be destabilized by Spillover(Ch.9, p.211)Integral control with state feedbackConstant disturbanceNon-zero steady state error on y Introduce the augmented state p such that : State feedback: Closed loop equation:If G and Gp are chosen so as to stabilize the system, without knowledge of the disturbance w(Ch.9, p.212)Frequency Shaped LQR (1)LQR:Parseval's theorem:Frequency independentFrequency-shaped LQR:To achieve P + I action At low frequency To increase the roll-off At high frequencyFrequency shaped LQR (2): weight specification P+I Increased roll-offFrequency shaped LQR (3): Augmented systemFrequency independent cost functionalState space realization of the augmented systemFrequency shaped LQR (4): Augmented systemThe state feedback of the augmented system is designed with the frequency independent Cost functional:Frequency shaped LQR (5): Architecture of the controllerOnly the states of the Structure must be reconstructedAugmented states(Ch.13, p.295)HAC / LAC strategy (1)The control system consists of tho imbedded loops: 1) The inner loop (LAC: Low Authority Control) consists of a decentralized active damping with collocated actuator/sensor pairs (no model necessary). The outer loop (HAC: High Authority Control) consists of a model-based non-collocated controller (based on a model of the actively damped structure).2)HAC / LAC strategy (2)Advantages: The active damping extends outside the bandwidth of the HAC (reduces the settling time of the modes beyond the bandwidth) The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the HAC loop (improved gain margin) The larger damping of the modes within the controller bandwidth makes them more robust to parametric uncertainty (improved phase margin)HAC / LAC strategy (3): Example Wide-band position control of a trussSet-upOpen-loop FRF of the HAC For various gains g of the LACHAC / LAC strategy (4): Example Wide-band position control of a trussBode plot of the controller HOpen-loop FRF of the design model: GHHAC / LAC strategy (5): Example Wide-band position control of a trussOpen-loop FRF of the full system: G*H Nyquist plotHigh frequency dynamicsStep response t (sec)`

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