Report

D u S -1 S S c1 c2 z1 … … z2 a1 a2 S S S S cn-1 cn zn-1 an-1 … S Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo System Solutions zn an y Outline of Today’s Lecture Review Convolution Equation Impulse Response Step Response Frequency Response Linearization Reachability Testing for Reachability Transformations We can transform our state space representation to other state variables (different that the ones in use). Mathematically, this is called a change of basis vectors. Why would we ant to do this? To make the problem easier to solve! To isolate a particular property of the system To uncouple the modes of the system Transformations Say we have some matrix T that is invertible (this is important) which results in the vector z when x is premultiplied by T. We then say that we have transformed the vector x into z, or alternatively, we have transformed x into z: x1 t1 1 z T x w here x ... , T is ... x n t n 1 xT 1 T hen T d d x 1 ... T 1 z1 exists, an d z ... z n z T ( Ax Bu ) TAx TBu TAT dt z Du L et A T A T d ... t1 n ... , t n n z dt y CT ... 1 z Az Bu dt y Cz Du , B TB , and C C T 1 . T hen 1 z T B u and Convolution Equation y (t ) C e A(t ) x (0 ) t Ce A ( t ) B u ( ) d D u ( t ) 0 is called the “Convolution Equation” Expresses the effect of an input on the system What is convolution? a twisting or folding together of two things A convolution is found in many phenomena: A sound that bounces off of a wall and interacts with the source sound is a convolution A shadow is a convolution between the light source and the object producing the shadow In statistics, a moving average is a convolution The Impulse Function Imagine a function that has a shape that is infinitesimally thin in the independent variable but infinitely high domain or response: In other words this is a very long and sharp spike This is what we try to model with the impulse function Mathematically we define the Dirac Delta Function, d(t), also called the Impulse Function by 0 1 u ( t ) p ( t ) 0 d ( t ) lim p ( t ) 0 t0 0t t System Response u (t ) d (t ) y (t ) C e y (t ) C e A(t ) x (0 ) t Ce x (0 ) A ( t ) t Ce A ( t ) B u ( ) d D u ( t ) 0 Bd (t )d D u (t ) C e A(t ) x (0 ) C e A(t ) B D u (t ) 0 If T hen A(t ) h (t ) y (t ) C e t Ce A ( t ) Bd (t )d C e A(t ) B 0 A(t ) x (0 ) t h ( t ) u ( ) d D u ( t ) 0 Since our system is linear and we can add solutions, we can approximate the response as a sum of the convolutions of h(t-)d(t) y(t) + + + + +… 12 345 t System Response S(t) A unit step is defined as 0 S (t ) 1 t0 1 t>0 t With zero initial conditions y (t ) t Ce A ( t ) B S ( ) d D S ( t ) 0 t C e A ( t ) Overshoot Mp Bd D 0 t C e A 0 1 Bd D C A e 1 A 1 CA e B CA B D At } } Transient Steady State B t 0 d { Steady State Rise time, tr Transient period=settling time, ts System Response Another common test function is a sinusoid for frequency response u ( t ) cos t e i t e i t 2 Since we have a linear system, we only need u (t ) e st where s i and assuming that the eigenvalues A do not equal s y ( t ) C e x (0 ) At t Ce s Be d D e At t e sI A At st At x (0 ) ( sI A ) 1 1 e sI A t I B D e st B C sI A 1 BD e st } } Ce s Be d D e 0 C e x (0 ) C e ( sI A ) At st 0 C e x (0 ) C e At A ( t ) Transient Steady State System Response: Frequency Response Time history with respect to a sinusoid: Phase Shift, DT Amplitude Ay Amplitude Au Input Sin(t) G ain Ay Au Period,T Transient Response P hase 2 DT T System Response Frequency Response y (t ) C e At x (0 ) ( sI A ) y ss ( t ) C sI A 1 1 B C sI A BD e st i Me e st 1 Me B D e i st M is the magnitude and is the phase G ain Ay M Au 1 D C G ain M 0 C A B D P hase 2 DT T y ss ( t ) M cos( t ) st Linearization Good solutions for the Linear Model Equally good techniques for the Nonlinear Model are not easy to come by What if the Nonlinear Model is well enough behaved in the region of interest so that we could apply Linear techniques strictly to that region? We did this with the inverted pendulum! We assumed small angles! Linearization Techniques Ignore the nonlinearity In some cases, the nonlinearity has a relatively small effect In those cases, build a linear system and treat the nonlinearity as a disturbance Small angle approximations sin cos 1 Often only useful near equilibrium points Taylor Series Truncation about an operating point 0 f ( x a ) f (a ) x df ( a ) dx 1 2 2 x 2 d f (a ) dx 2 ... Assumes that 2nd and higher orders are negligible Feedback linearization Reachability Consider the following problem: With the linkage below, can you control the position of p? y y’ yp x’ u(t) x p Reachability We define reachability (often times called controllability) by the following: A state in a system is reachable if for any valid states of the system, say, initial state at time t=0, x0 , and a state xf , there exists a solution for t>0 such that x(0) = x0 and x(t)=xf. There are systems which we can not control the states are not reachable with our input. There in designing control systems, it is important to know if the system is controllable. This is closely linked with the concept of ergodicity of the system in which we ask the question whether or not it is possible to with some measure of our system to measure every possible state of the system. Reachability Testing d x Ax Bu and y C x D u w here x (0 ) 0 and D 0 dt x (t ) t e A ( t ) 0 B u d Im pulse response is x d t e A ( t ) 0 B d d e B At T he response to the rate of change in th e im pulse response is xd dx d Ae B At dt for u ( t ) 1d t d t , the response w ould then be x ( t ) 1 e B 2 A e B At C ontinuing the process through further d erivatives, w e get for u ( t ) 1d t 2 d t 3 d t 4 d ( n 1) t ... n d t x ( t ) 1 e B 2 A e B 3 A e B 4 A e B ... n A At At 2 At 3 At lim x ( t ) 1 B 2 A B 3 A B 4 A B ... n A 2 t 0 lim x ( t ) B t 0 W r B AB AB A A n 1 n 1 3 n 1 n 1 B B B is called the reachability m atrix At e B At Reachability For the system, d , x Ax Bu and y C x D u dt all of the states of the system are reachable if and only if Wr is invertible where Wr is given by W r B AB A n 1 B State Space Formulation T o p u t it in th e d esired fo rm Is this a reachable system? L et z1 z , z 2 z , z 3 z u , z4 zu T h en w e can w rite z1 z 2 z3 z4 m z bz kz bz u kz u m u z u bz u ( k k t ) z u k t z r bz kz m z bz kz bz u kz u m u z u bz u ( k k t ) z u k t z r bz kz T he state variables are z, z, zu , zu u z r is the input airfield profile T he output i s z, the nose deflection z2 z4 0 z1 k m d z2 = dt z3 0 z4 k m u kz1 m kz1 m mu bz 2 k m m 0 0 b mu bz 4 y 1 0 0 z1 z2 0 z3 z4 bz 4 mu kt zr mu 0 z b 1 0 m z2 0 u 1 z3 kt b z4 m u m u m 5, 000, 00 0 T m (k k t ) z3 0 k kt mu kz 3 mu 0 b m mu 1 bz 2 mu 50 k 250, 000 k t 1, 250, 000 b 125, 000 Example z1 0 0.05 d z2 = dt z3 0 z 4 5000 1 0 0.025 0.05 0 0 2500 30000 0.05 125 2 A 5000 7 1.25 * 10 W r B AB 0.025 0.05 62.4 5 750 2500 30000 6.245 * 10 125 312300 3 A 7 1.25 * 10 10 3.11 * 10 156000 6.245 * 10 1.554 * 10 A B 6 7.5 * 10 62.45 2500 6 6.22 * 10 7 62.45 1.874 * 10 6 10 0 625 AB 25000 7 6.25 * 10 7.5 * 10 6 7 1.866 * 10 0 0 3 A B 0 25000 625 6 1.562 * 10 2 A B 7 6.25 * 10 11 1.555 * 10 0.025 750 62.45 2 z1 0 z 0.025 0 2 u 1 z3 0 2500 z 4 25000 0 11 1.561 * 10 6 9 3.884 * 10 2 A B 11 1.555 * 10 14 3.869 * 10 155400 6 6.22 * 10 10 1.548 * 10 0 625 625 1.562 * 10 25000 6.25 * 10 6.25 * 10 7 1.555 * 10 6 7 11 6 1.561 * 10 9 3.884 * 10 11 1.555 * 10 14 3.869 * 10 Example W r B Wr 1 AB 0.9958 23.75 1.975 0.00079 2 A B 0 0 3 A B 0 25000 23.75 0.4938 10 0.25 0.9992 0.02498 0. 0004 0.00001 0 625 625 1.562 * 10 25000 6.25 * 10 6.25 * 10 0.00004 0 0 0 7 1.555 * 10 6 7 11 6 1.561 * 10 9 3.884 * 10 11 1.555 * 10 14 3.869 * 10 T he inverse of the reachability m a trix exists Since the inverse of the reachability matrix exists, the system is reachable and controllable. Matlab Matlab has the function ctrb(A,B) which will compute the reachability matrix: >> A=[0 1 0 0; -0.05 -0.025 0.05 0.025;... 0 0 0 1; 5000 2500 -30000 -2500] A= 1.0e+004 * 0 0.0001 0 0 -0.0000 -0.0000 0.0000 0.0000 0 0 0 0.0001 0.5000 0.2500 -3.0000 -0.2500 >> B = [0;0;0;25000] B= 0 0 0 25000 >> Wr=ctrb(A,B) Wr = 1.0e+014 * 0 0 0.0000 -0.0000 0 0.0000 -0.0000 0.0000 0 0.0000 -0.0000 0.0016 0.0000 -0.0000 0.0016 -3.8688 >> inv(Wr) Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 9.669679e-017. ans = 0.9937 23.7500 -0.4938 0.0000 23.7500 -10.0000 0.2500 0 1.9750 -0.9992 0.0250 0 0.0008 -0.0004 0.0000 0 Canonical Forms The word “canonical” means prescribed In Control Theory there a number transformations that can be made to put a system into a certain canonical form where the structure of the system is readily recognized One such form is the Controllable or Reachable Canonical form. Reachable Canonical Form A system is in the reachable canonical form if it has the structure z1 a 1 z2 1 d z3 0 dt ... ... z n 0 y b1 b2 b3 a2 a3 ... 0 0 ... 1 0 ... 0 0 1 a n z1 1 z 0 0 2 0 z3 0 u ... ... 0 0 z n 0 bn z D u ... Such a structure can be represented by blocks as D u S -1 S S c1 c2 z1 … … z2 a1 a2 S S S S cn-1 cn zn-1 an-1 … S zn an y Reachable Canonical Form It can be shown that the characteristic polynomial is A I a1 n n 1 a2 n2 ... a n 1 a n 2 To convert to Reachable Canonical Form, consider the transformation A TA T W r 1 B TB n 1 B AB A B TA T 1 A B TB TA B 2 A B ( TA T n ... 1 ) TB TA T 2 1 TA T 1 TB TA B 2 A B TA B W r n T B AB ... A n 1 B TW r T W rW r 1 Reachability Canonical Form a1 1 A 0 ... 0 W r B a2 a3 ... 1 an 0 0 ... 0 0 0 ... 1 0 ... A ... 0 0 AB A B n 1 2 1 0 B 0 0 0 C CT 1 B for a 4 state variable state m atrix, W r 1 0 0 0 a1 a1 a 2 1 a1 0 1 0 0 a1 2 a1a 2 a 3 2 a1 a 2 a1 1 2 3 w here the a i are the coefficients of the c haracteristic polyn o m ia l I A a1 a 2 a 3 a 4 4 3 2 Example: Inverted Pendulum Develop the reachability canonical form for the Segway using the inverted pendulum model of Lecture 5 0 0 A 0 0 1 0 0 6.405 0 0 0 7.205 0 0 1 0 0 0.01837 B 0 0.008163 T he eigenvalues of A are {2.684, 2.684, 0., 0.} 0 0 AB 0 0 U sing the m odel developed in L ecture 5 for the inverted pendulum 0 x 0 d v dt 0 0 y 0 1 1 0 2 2 0 m l g J (M m) Mml 0 0 0 0 m lg ( M m ) J (M m) Mml x v 0 2 2 0 0 x J ml2 0 2 v J (M m) Mml 1 0 ml 0 2 J ( M m ) M m l M 10 kg w ith m 80 kg l 1m J 100 kg m / s 2 2 F 0 0 2 A B 0 0 1 0 0 6.405 0 0 0 7.205 1 0 0 6.405 0 0 0 7.205 0.05228 0. 3 A B 0.05882 0. 0 0 0.01837 0 0.01837 0. 1 0 0.008163 0 0.00816 3 0. 0 0 0 0 1 0 0 0 1 0 0 6.405 0 0 0 7.205 0 0 0. 0 0.01837 0.05228 1 0 0. 0 0.008163 0.05882 Example W r B A B 0 0.01837 Wr 0 0.0081633 T W rW r A TA T 1 0 1. A 0 0 C CT 1 1 1 0 0 0 0. 0. 0.05228 0.008163 0. 0. 0.05 882 0 7.2051 1. 0 0. 1. 0. 0. 7.205 0 0 0 1. 0 0 1. z1 0 1. d z2 dz z 3 0 z4 0 4 0.01837 0 0 0 12.491 0 T he characteristic polynom ial of A is I A 7.205 3 A B 2 AB 1 0 a1 a1 a 2 1 a1 0 1 0 0 0. 7.2051 90. 0 0. 1. 12.491 90. 0. 0. 80. 12.491 0. 0. 28.10 80. 0 0. 0 28.10 0 0. 12.491 0 0 0 0 122.5 12.491 0 0. 28.10 0 0 0 0 1 0 0 0 0.05228 0. 0.05882 0. 122.5 0 0 0 28.10 0 0. 0 0 0 B TB 0 12.491 0 0.01837 0 0 0.008163 7.205 0 0 0 1. 0 0 1. 0 z1 0 z2 0 z3 0 z4 W 1 0 0 6.405 0 0 0 7. 205 0 0 122.5 12.491 0 0. 28.10 0 0 0.0801 0.008163 0 0 0 y 0 0 0 0 0.01837 1 0 0 0.008163 0 0.01837 1 0 u 0 0 r 0.008163 2 0 0 0 122.5 12.491 0 0. 28.10 0.01837 0 0 0.0801 0.008163 0 0 0 122.5 0 1 0 0.01837 0 28 .10 0 0 0 . 0.008163 0 0.08001 0 0 0 0 0 3 a1 2 a1a 2 a 3 1 2 0 a1 a 2 0 a1 1 0 0.008163 z1 z2 0.008163 z3 z4 0 0.008163 0 7.2051 1. 0 0. 1. 0. 0. 122.5 0 28.10 0. 0.08001 0 0 0 2 7.2051 0 1. 0 Summary Reachability A state in a system is reachable if for any valid states of the system, say, initial state at time t=0, x0 , and a state xf , there exists a solution for t>0 such that x(0) = x0 and x(t)=xf. Testing for Reachability W r B AB For the system, A d n 1 B is called the reachability m atrix x Ax Bu and y C x D u , all of the states of the system are reachable if and only if Wr is invertible where Wr is given by dt W r B AB A n 1 Next: State Feedback B