2. At this point, we can analyze (stability, steady-state gain, sinusoidal steady-state gains, time-constant, etc.) of first-order, linear dynamical systems. We previously analyzed a Ist-order process model, and a proportional-control strategy. In this problem, we try a different situation, where the process is simply proportional, but the controller is a Ist-order, linear dynamical system. Specifically, suppose the process model is non-dynamic ("static" ) simply y(t) = cu(t) + Bd(t) where o and B are constants. The control strategy is dynamic i (t) = ar(t) + bir(t) + bzym(t) u(t) = cr(t) + dir(t) where ym(t) = y(t) + n(t) and the various "gains" (a, bi, . .., di) constitute the design choices in the control strategy. Be careful, notation-wise, since (for example) d, is a constant parameter, and d(t) is a signal (the disturbance). (a) Eliminate u and ym from the equations to obtain a differential equation for r of the form r(t) = Ar(t) + Bir(t) + Bad(t) + Ban(t) which governs the closed-loop behavior of r. Note that A, B1, B2, By are functions of the parameters a, b1, ... in the control strategy, as well as the process parameters o and B. (b) What relations on (a, b1. .... dj, or, B) are equivalent to closed-loop system stability? (c) As usual, we are interested in the effect (with feedback in place) of (r, d, n) on (y, u), the regulated variable, and the control variable, respectively. Find the coefficients (in terms of (a, bi, . . ., d1, 0, B)) so that y(t) = Cix(t) + Dur(t) + Died(t) + Dian(t) u(t) = Car(t) + Dar(t) + Dad(t) + Dzan(t) (d) Suppose that T. > 0 is a desired closed-loop time constant. Write down the constraints on the a, b1, b2, c and di (i.e., the parameters of the controller to be design) such that the following conditions hold: . closed-loop is stable . closed-loop time constant is To . steady-state gain from d -> y is 0 . steady-state gain from r - y is 12. At this point, we can analyze (stability, steady-state gain, sinusoidal steady-state gains, time-constant, etc.) of first-order, linear dynamical systems. We previously analyzed a Ist-order process model, and a proportional-control strategy. In this problem, we try a different situation, where the process is simply proportional, but the controller is a Ist-order, linear dynamical system. Specifically, suppose the process model is non-dynamic ("static" ) simply y(t) = cu(t) + Bd(t) where o and B are constants. The control strategy is dynamic i (t) = ar(t) + bir(t) + bzym(t) u(t) = cr(t) + dir(t) where ym(t) = y(t) + n(t) and the various "gains" (a, bi, . .., di) constitute the design choices in the control strategy. Be careful, notation-wise, since (for example) d, is a constant parameter, and d(t) is a signal (the disturbance). (a) Eliminate u and ym from the equations to obtain a differential equation for r of the form r(t) = Ar(t) + Bir(t) + Bad(t) + Ban(t) which governs the closed-loop behavior of r. Note that A, B1, B2, By are functions of the parameters a, b1, ... in the control strategy, as well as the process parameters o and B. (b) What relations on (a, b1. .... dj, or, B) are equivalent to closed-loop system stability? (c) As usual, we are interested in the effect (with feedback in place) of (r, d, n) on (y, u), the regulated variable, and the control variable, respectively. Find the coefficients (in terms of (a, bi, . . ., d1, 0, B)) so that y(t) = Cix(t) + Dur(t) + Died(t) + Dian(t) u(t) = Car(t) + Dar(t) + Dad(t) + Dzan(t) (d) Suppose that T. > 0 is a desired closed-loop time constant. Write down the constraints on the a, b1, b2, c and di (i.e., the parameters of the controller to be design) such that the following conditions hold: . closed-loop is stable . closed-loop time constant is To . steady-state gain from d -> y is 0 . steady-state gain from r - y is 11. We use the added variable technique to derive the variance ination factor (VIP). Consider a linear model of the form 91' =50+l31$1+l3213922+-"+}3p$a'p+zr, 5'3: 1:"'ana (1) where the errors are uncorrelated with mean zero and variance 02. Let X denote the n X p' predictor matrix and assume X is of full rank. We will derive the VIP for ip. The same derivation applies to any other coefcient simply by rearranging the columns of X. Let U denote the matrix containing the rst p' 1 columns of X and let z denote the the last column of X so that X = [U 2]. Then we can write the model in (1) as 50 x91 Y=[U z](,:J)+t-:=Ua+z6p+e with a: (2) x810. 1 Let 2 denote the vector of tted values from the least squares regression of z on the columns of U (Le. the regression of X.p on all the other variables), and let T : z 2 denote the residuals from that regression. Note that 'r' and 3 are not random, they are constant vectors obtained by linear transformations of z. (a) Show that the regression model in (2) can be rewritten in the form for some constant vector 6 of the same length as a. (Hint: z : i l 'r and 2? = U(UTU)_1UTz). (b) Show that UT? 2 0, a zero vector. (0) Obtain simplied expressions for the least squares estimators of 5 and 5?, showing, in particular, that 5,, : 'rTY/rT'r. (d) Based on Part (c) and the model assumptions, show that 0.2 ELK\"? _ is)? where :Eg-p is the LS tted value from regression X,D on the all the other predictor variables with an intercept. var(,p) : Problem 15 Labour Demand with Perfect Competition in the Labour Market and Perfect Competition in the Output Market in Short Run. You are the manager of a business that operates in perfectly competitive markets (both the Labour Market and Output Market). The production function of the business is given by: Q -40LU . The price of the product is 100. The wage rate is 20. a. Calculate the level of output and amount of labour hired by the firm. b. Derive the demand for labour (Hint: solve for L in terms of w). c. Calculate the elasticity of Labour Demand and interpret the result in WORDS. d. Make diagrams for your solutions