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**Lecture 15: State Feedback Control: Part I**

Pole Placement for SISO Systems Illustrative Examples

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**Feedback Control Objective**

In most applications the objective of a control system is to regulate or track the system output by adjusting its input subject to physical limitations. There are two distinct ways by which this objective can be achieved: open-loop and closed-loop.

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**Open-Loop vs. Closed-Loop Control**

Open-Loop Control Output Desired Output Command Input Controller System Closed-Loop Control Output Desired Output Controller System - Sensor

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**Closed-Loop Control Advantages Disadvantages**

automatically adjusts input less sensitive to system variation and disturbances can stabilize unstable systems Disadvantages Complexity Instability

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**State Feedback Control Block Diagram**

u x y H Z-1 C G -K

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**State-Feedback Control Objectives**

Regulation: Force state x to equilibrium state (usually 0) with a desirable dynamic response. Tracking: Force the output of the system y to tracks a given desired output yd with a desirable dynamic response.

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Closed-loop System Plant: Control: Closed-loop System:

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**Pole Placement Problem**

Choose the state feedback gain to place the poles of the closed-loop system, i.e., At specified locations

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**State Feedback Control of a System in CCF**

Consider a SISO system in CCF: State Feedback Control

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**Closed-Loop CCF System**

Closed loop A matrix:

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**Choosing the Gain-CCF Closed-loop Characteristic Equation**

Desired Characteristic Equation: Control Gains:

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**Transformation to CCF To CCF Transform system**

Where x+(k)=x(k+1) (for simplicity) First, find how new state z1 is related to x:

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**Transformed State Equations**

Necessary Conditions for p: 1 Vector p can be found if the system is controllable:

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**State Transformation Invertibility**

Matrix T is invertible since By the Cayley-Hamilton theorem.

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Toeplitz Matrix The Cayley-Hamilton theorem can further be used to show that Matrix on the right is called Toeplitz matrix

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**State Transformation Formulas**

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**State Feedback Control Gain Selection**

By Cayley Hamilton: or

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Bass-Gura Formula

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**Double Integrator-Matlab Solution**

T=0.5; lam=[0;0]; G=[1 T;0 1]; H=[T^2/2;T]; C=[1 0]; K=acker(G,H,lam); Gcl=G-H*K; clsys=ss(Gcl,H,C,0,T); step(clsys);

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**Flexible System Example**

Consider the linear mass-spring system shown below: x1 x2 Parameters: m1=m2=1Kg. K=50 N/m k u m1 m2 Analyze PD controller based on a)x1, b)x2 Design state feedback controller, place poles at

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Collocated Control Transfer Function: PD Control: Root-Locus

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**Non-Collocated Control**

Transfer Function: PD Control: Root-Locus Unstable

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State Model Discretized Model: x(k+1)=Gx(k)+Hu(k)

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**Open-Loop System Information**

Controllability matrix: Characteristic equation: |zI-G|=(z-1)2(z2-1.99z+1)=z4-3.99z3+5.98z2-3.99z+6

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**State Feedback Controller**

Characteristic Equations: |zI-G|=(z-1)2(z2-1.99z+1)=z4-3.99z3+5.98z2-3.99z+6

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**Matlab Solution %System Matrices m1=1; m2=1; k=50; T=0.01;**

syst=ss(A,B,C,D); A=[ ; ; ; ]; B=[0; 0; 1; 0]; C=[ ; ]; D=zeros(2,1); cplant=ss(A,B,C,D); %Discrete-Time Plant plant=c2d(cplant,T); [G,H,C,D]=ssdata(plant);

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**Matlab Solution %Desired Close-Loop Poles**

pc=[-20;-20;-5*sqrt(2)*(1+j);-5*sqrt(2)*(1-j)]; pd=exp(T*pc); % State Feedback Controller K=acker(G,H,pd); %Closed-Loop System clsys=ss(G-H*K,H,C,0,T); grid step(clsys,1)

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Time Respone

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**Steady-State Gain Closed-loop system: x(k+1)=Gclx(k)+Hr(k), Y=Cx(k)**

Y(z)=C(zI-Gcl)-1H R(z) If r(k)=r.1(k) then yss=C(I-Gcl)-1H Thus if the desired output is constant r=yd/gain, gain= C(I-Gcl)-1H

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Time Response

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**Integral Control Control law: 1/g yd u x y Ki H C G plant**

Z-1 C G plant Integral controller -Ks Automatically generates reference input r!

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**Closed-Loop Integral Control System**

Plant: Control: Integral state: Closed-loop system

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**Double Integrator-Matlab Solution**

T=0.5; lam=[0;0;0]; G=[1 T;0 1]; H=[T^2/2;T]; C=[1 0]; Gbar=[G zeros(2,1);C 1]; Hbar=[H;0]; K=acker(Gbar,Hbar,lam); Gcl=Gbar-Hbar*K; yd=1; r=0; %unknown gain clsys=ss(Gcl,[H*r;-yd],[C 0;K],0,T); step(clsys);

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**Closed-Loop Step Response**

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