Control Problems: A . . . Resulting Problem Analysis of the Problem Cost of Actuators: . . . What Is the Economically Cost of Measurements Optimal Way to Guarantee Overall Cost Resulting . . . Interval Bounds on Control? How to Solve This . . . General Case Alfredo Vaccaro 1 , Martine Ceberio 2 , and Home Page Vladik Kreinovich 2 Title Page 1 Department of Engineering ◭◭ ◮◮ University of Sannio, Italy 2 Department of Computer Science ◭ ◮ University of Texas at El Paso, USA contact email vladik@utep.edu Page 1 of 34 Go Back Full Screen Close Quit
Control Problems: A . . . Resulting Problem 1. Control Problems: A Very Brief Reminder Analysis of the Problem • In control problems, we need to find the values of the Cost of Actuators: . . . control u = ( u 1 , . . . , u n ) . Cost of Measurements Overall Cost • Usually, there are some requirements on the control. Resulting . . . • For example, we may require that under this control, How to Solve This . . . the system should be stable. General Case Home Page • These conditions are usually described by inequalities. Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 34 Go Back Full Screen Close Quit
Control Problems: A . . . Resulting Problem 2. From Optimal Control to Constraint Satisfac- Analysis of the Problem tion Cost of Actuators: . . . • In general, there are many different controls that sat- Cost of Measurements isfy all the desired constraints. Overall Cost Resulting . . . • In the ideal case: How to Solve This . . . – we know the exact initial state of the system and General Case – we know the equations that describe the system’s Home Page dynamics under different controls. Title Page • Then, we can compute the exact consequences of each ◭◭ ◮◮ control. ◭ ◮ • Thus, depending on what is our objective, we can: Page 3 of 34 – select an appropriate objective functions and Go Back – look for the control that optimized this objective function. Full Screen • The objective function depends on the task. Close Quit
Control Problems: A . . . Resulting Problem 3. From Optimal Control to Constraint Satisfac- Analysis of the Problem tion (cont-d) Cost of Actuators: . . . • For example, for selecting a plane’s trajectory, we can Cost of Measurements have different objective functions. Overall Cost Resulting . . . • In the situation of medical emergency: How to Solve This . . . – we need to find the trajectory of the plane General Case – that brings the medical team to the remote patient Home Page as soon as possible. Title Page • For a regular passenger communications: ◭◭ ◮◮ – we need to minimize expenses ◭ ◮ – so, we should fly at the speed that saves as much Page 4 of 34 fuel as possible. Go Back • For a private jet, a reasonable objective function is the Full Screen ride’s smoothness. Close Quit
Control Problems: A . . . Resulting Problem 4. From Optimal Control to Constraint Satisfac- Analysis of the Problem tion (cont-d) Cost of Actuators: . . . • In practice, we rarely know the exact initial state and Cost of Measurements the exact system’s dynamics. Overall Cost Resulting . . . • Often, for each of the corresponding parameters, we How to Solve This . . . only know the lower and upper bounds on values. General Case • In other words, we only know the interval that contains Home Page all possible values of the corresponding parameter. Title Page • In such cases, for each control: ◭◭ ◮◮ – instead of the exact value of the objective function, ◭ ◮ – we get the range of values: [ v, v ] . Page 5 of 34 Go Back Full Screen Close Quit
Control Problems: A . . . Resulting Problem 5. From Optimal Control to Constraint Satisfac- Analysis of the Problem tion (cont-d) Cost of Actuators: . . . • Computing this range is a particular case of the main Cost of Measurements problem of interval computations . Overall Cost Resulting . . . • In such situations, it make sense, e.g., to describe all How to Solve This . . . the controls u which are possibly optimal. General Case • In other words, we want to describe all the controls for Home Page which Title Page v ( u ′ ) . v ( u ) ≥ max u ′ ◭◭ ◮◮ ◭ ◮ Page 6 of 34 Go Back Full Screen Close Quit
Control Problems: A . . . Resulting Problem 6. Resulting Problem Analysis of the Problem • In situations of interval uncertainty, interval methods Cost of Actuators: . . . enable us: Cost of Measurements Overall Cost – to find a box B = [ u 1 , u 1 ] × . . . [ u n , u n ] Resulting . . . – for which any control u ∈ B has the desired prop- How to Solve This . . . erties – such as stability or possible optimality. General Case • Thus, in real-life control, we need to make sure that Home Page u i ∈ [ u i , u i ] for all parameters u i describing control. Title Page • What is the most economical way to guarantee these ◭◭ ◮◮ bounds? ◭ ◮ Page 7 of 34 Go Back Full Screen Close Quit
Control Problems: A . . . Resulting Problem 7. Analysis of the Problem Analysis of the Problem • Actuators are never precise, so we can only set up the Cost of Actuators: . . . control value u i with some accuracy a i . Cost of Measurements = u i + u i Overall Cost def • Thus, if we aim for the midpoint u mi , we 2 Resulting . . . will get the actual value u i within the interval How to Solve This . . . [ u mi − a i , u mi + a i ] . General Case Home Page • The only way to guarantee that the control value is Title Page indeed within the desired interval is to measure it. ◭◭ ◮◮ • Measurement are also never absolutely precise. ◭ ◮ • Let us assume that we use a measuring instrument with accuracy ε i ; this means that: Page 8 of 34 – for each actual value u i of the corresponding pa- Go Back rameter, Full Screen – the measured value � u i is somewhere within the in- Close terval [ u i − ε i , u i + ε i ] . Quit
Control Problems: A . . . Resulting Problem 8. Analysis of the Problem (cont-d) Analysis of the Problem • Based on the measurements: Cost of Actuators: . . . Cost of Measurements – the only thing we can conclude about the actual Overall Cost (unknown) value u i Resulting . . . – is that it belongs to the interval [ � u i + ε i ] . u i − ε i , � How to Solve This . . . • We want to make sure that all the values from this General Case interval are within the desired interval [ u i , u i ] , i.e., that Home Page u i − ε i and � u i + ε i ≤ u i . u i ≤ � Title Page ◭◭ ◮◮ • These inequalities must hold for all possible values ◭ ◮ u i ∈ [ u i − ε i , u i + ε i ] . � Page 9 of 34 • We want the inequality u i ≤ � u i − ε i to hold for all these Go Back values. Full Screen • It is sufficient to require that this inequality holds for the smallest possible value � u i = u i − ε i . Close Quit
Control Problems: A . . . Resulting Problem 9. Analysis of the Problem (cont-d) Analysis of the Problem • So, u i ≤ ( u i − ε i ) − ε i = u i − 2 ε i . Cost of Actuators: . . . Cost of Measurements • Similarly, the inequality � u i + ε i ≤ u i should hold for all Overall Cost the values � u i ∈ [ u i − ε i , u i + ε i ] . Resulting . . . • It is sufficient to require that this inequality holds for How to Solve This . . . the largest possible value General Case Home Page u i = u i + ε i . � Title Page • So, ( u i + ε i ) + ε i = u i + 2 ε i ≤ u i . ◭◭ ◮◮ • The inequalities u i ≤ u i − 2 ε i and u i + 2 ε i ≤ u i must ◭ ◮ hold for all possible values u i ∈ [ u mi − a i , u mi + a i ]. Page 10 of 34 • For the 1st inequality, it is sufficient to require that it Go Back holds for the smallest possible value u i = u mi − a i . Full Screen • So, u i ≤ u mi − 2 ε i − a i . Close Quit
Control Problems: A . . . Resulting Problem 10. Analysis of the Problem (cont-d) Analysis of the Problem • Similarly, for the 2nd inequality, it is sufficient to re- Cost of Actuators: . . . quire that it holds for the largest possible value Cost of Measurements Overall Cost u i = u mi + a i . Resulting . . . How to Solve This . . . • So, u mi + 2 ε i + a i ≤ u i . General Case • Let us denote the half-width of the interval [ u i , u i ] by Home Page = u i − u i def Title Page ∆ i . 2 ◭◭ ◮◮ • In terms of the half-width, u i − u mi = u mi − u i = ∆ i . ◭ ◮ • Thus, the above inequalities are equivalent to the in- Page 11 of 34 equality 2 ε i + a i ≤ ∆ i . Go Back Full Screen Close Quit
Control Problems: A . . . Resulting Problem 11. In the Optimal Solution, We Have Equality Analysis of the Problem • In general: Cost of Actuators: . . . Cost of Measurements – the more accuracy we want, Overall Cost – the more expensive will be the corresponding mea- Resulting . . . surements and actuators. How to Solve This . . . • From this viewpoint, if 2 ε i + a i < ∆ i , then: General Case Home Page – we can use slightly less accurate actuators and/or measuring instruments Title Page – and still guarantee the desired inequality. ◭◭ ◮◮ • Thus, in the most economical solution, in the corre- ◭ ◮ sponding formula, we should have the exact equality: Page 12 of 34 2 ε i + a i = ∆ i . Go Back Full Screen • This is equivalent to a i = ∆ i − 2 ε i . Close Quit
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